Hi, I hope that you do have some degree of stability in your symptoms as you await solutions, and that you have people around you for support.
My thoughts, and one possible solution resulting from a PubMed crawl
I haven’t had surgery, I do not have EDS, nor am I familiar with the condition aside from a glance at a search result and noting that with EDS, the connective tissues in blood vessels are not as “strong and elastic”.
Just regarding:
I could dig through PubMed for information on EDS and how that affects vasculature for you, if reading from a screen is not a huge problem in your state.
The A&P text that I usually refer to first has an incomplete but simplified diagram of the venous outflow of the head & neck:
I don’t recall the specifics of what’s incomplete, I’d have to dig through for recall.
An image search for “venous drainage of the head, neck and brain” could be interesting.
Regarding EDS connection, there’s a phenomenon called “collateralization” in vascular science or whatever. Tissues and their cells that make up the arteries and veins are kind of specialized to be really good at finding a way out when things change or whatever. Scans could reveal where your veins have expanded to compensate, or show that your veins are having trouble expanding, or I suppose where/how the ligaments or whatever connections between your skull and atlas (as per my brief glance at a Wikipedia page on CCI) or whatever are interfering with your veins. Conceivably, some way to move or some way not to move or whatever might be inferrable from this. My thoughts were isometric neck exercises, to sort of get the veins to get their shit together in “use it or lose it” fashion, but obviously, I don’t have lived experience with EDS. “Are the headaches responsive to light cardio” generally, is what I would wonder, and I’d maybe check YouTube or whatever for “EDS headache exercises” and see what comes up. Obviously, being very judicious and perhaps consulting with someone nearby or here.
I missed this part in the above reply. This is a stumper for me off the cuff… in terms of why this is the case, before even getting into what that means. What I do know is that the nerves, even while innervating blood vessels, are themselves surrounded by like a plexus of microvasculature, the details of which elude me. I looked it up only very recently. In trying to understand how EDS relates to the smaller nerves, I would wonder whether the smaller nerve caliber is due to the smaller microvasculature exclusively, or if maybe the smaller microvasculature is unable to nourish any wider nerves, or if the nerve tissue itself consists of significant amounts of connective tissue which for whatever reason also is supposed to expand in some capacity perhaps by whatever microscopic amounts of fluid involved in nerve cell function?? 
EDIT:
Also, just supplementing with a bit of salient A&P 101: all the hundreds? thousands? of types of tissues (basically cell lattices) in the body are categorized into just four buckets: muscle tissue, nerve tissue, epithelial tissue, and connective tissue. I’m not sure that I’ve encountered a real good philosophical reason for the last bucket “connective tissue”, which like randomly includes platelets for whatever reason and such. So finding out what tissue specifically is affected by EDS would be worth a look, so as to narrow the scope of consideration.
EDIT:
Also, not sure what “not strong” means, but I would assume this means “liable to thrombosis” or tearing or whatever. “Not elastic” maybe means higher blood pressure or something, not being able to expand as well.
Maybe even a visit to the ER or something for any acute vascular damage. People who don’t complain much maybe tend to over-suffer…
EDIT:
If you happen to be into meditation, I’ve got lots to say about that, but that could get complicated… Maybe find some way to ‘activate the relaxation response’ or whatever for vascular dilation or whatever… Maybe like Tai Chi or Qi Gong, which recently had some results for hypertension, with a mediating mechanism unknown… 
But yeah, I hope you get what you need…
EDIT:
A perusal of PubMed through the research paper posted recently about the connection between intracranial hypertension and EDS:
Using the similar articles feature:
Similar articles for PMID: 38269002 - Search Results - PubMed
Here’s a salient title from that list:
Connective tissue, Ehlers-Danlos syndrome(s), and head and cervical pain - PubMed
It’s from 2015 and the abstract begins with:
Ehlers-Danlos syndrome (EDS) is an umbrella term for a growing group of hereditary disorders of the connective tissue mainly manifesting with generalized joint hypermobility, skin hyperextensibility, and vascular and internal organ fragility. In contrast with other well known heritable connective tissue disorders with severe cardiovascular involvement (e.g., Marfan syndrome), most EDS patients share a nearly normal life span, but are severely limited by disabling features, such as pain, fatigue and headache.
Limited by institutional or ‘gray hat’ access.
Also, from the first EDS-IIH article, perusing the references, I saw the title “Can Elevated IGF‐1 levels among patients With Ehlers–Danlos syndrome cause idiopathic intracranial hypertension?”. It’s citation number 14, so searching the article for “(14)” I see that the reference is for the sentence:
While the physiology underlying the unique linkage between EDS and IIH is still widely unknown, abnormal levels of growth factors, such as IGF-1, have been proposed as one possible cause (14).
The abstract of the article says:
We offer for consideration a possible association between hypermobility syndrome seen in Ehlers–Danlos syndrome and risk of potential development of idiopathic intracranial hypertension – mediated primarily through the effects of insulin-like growth factor-1.
Again, not readily accessible, but through institutional or ‘gray hat’ means. The article is from 2013/2015. There were some slew of papers on IIH ever since, but given that the referring paper is from just this year, I guess it’s not completely invalidated.
The intro says:
Idiopathic intracranial hypertension (IIH) – formerly known as “pseudotumor cerebri” – is a neurological disorder characterized by an increase in intracranial pressure in the absence of a brain tumor or other secondary disease process.The pathophysiology of IIH is controversial: studies have identified the significance of growth hormone (GH) as well as insulin-like growth factor-1 (IGF-1) in mediating the increase in pressure – although the mechanism behind this process is not yet clearly understood. Here, we offer for consideration a possible association between hypermobility syndrome (HMS) seen in Ehlers–Danlos syndrome and risk of potential development of IIH – mediated primarily through the effects of IGF-1.
Notably, GH and IGF-1 are produced by the anterior pituitary gland and the “liver and other tissues”, respectively, according to the A&P text here:
The article continues with a case report:
In June 2010, a 36-year-old non-obese female (body mass index [BMI] 18.5) presented with symptoms of a generalized headache for 8 months. She described the headaches as throbbing, worse in the morning hours, disabling, and exacerbated by recumbency. Moreover, the patient reported that the headaches were similar in nature to those she had experienced as a teenager following a lumbar puncture (LP) performed to investigate visual obscurations following an auto accident. With the current headaches, the patient reported experiencing some nausea as well as occasional double vision. She denied having any sensitivity to light, sound, or smell; she also denied having any symptoms of lacrimation, rhinorrhea, or conjunctival injection associated with her headaches.
…
… In February 2011, we started therapy with subcutaneous octreotide, based on several reports from the medical literature. The patient had a dramatic clinical response to this treatment – with elimination of her visual symptoms and head pain. …
So it’s this one case of a patient that they were doing lumbar punctures on multiple times to alleviate symptoms, when they finally somehow incidentally figured out something that worked. Wikipedia says that octreotide is a growth hormone inhibitor. Maybe worth bringing up with a GP or something…?
A blood test to confirm elevated IGF-1, and an injection of octreotide if called for, basically. Obviously, close reading advised. I’m not at all a medical professional.
The discussion section talks about why it might have worked. I’ll save that and a discussion of the “similar” paper for tomorrow.
EDIT:
Specifically, there has been evidence suggesting that IGF-1 receptors present at the level of the choroid plexus – the site of CSF production – are excessively stimulated in high GH-IGF-1 states (as seen in cases of acromegaly or HMS) leading to excess CSF production and thus increased intracranial pressure.
The choroid plexus, last I looked it up, was somewhere in the ventricles. They’re responsible for CSF production at like some really slow rate. CSF is basically like filtered through ‘the blood-brain-barrier’, though I forget what the general distribution of that barrier tissue is like… How that all comes together is on my shortlist of things to look into…
eg:
In any case, do note that I’d be quite happy for this comment to be deleted after whatever useful information is gleaned, perhaps more carefully presented.