I think @Isaiah_40_31 has said pretty much all I’d be able to contribute…I do agree with you that you have had plenty of testing & obviously your CT shows the styloid & calcified ligament! MRIs don’t use radiation so that’s not an issue, but ones with contrast usually use gadolinium as the contrast medium which can cause side effects, so might be an idea to read up on this before you make a decision- we’ve had some info posted on here by @vdm in this discussion:
MRI contrast risks / Gadolinium toxicity - General - Living with Eagle
A CT does expose you to more radiation, but uses a different contrast medium & doesn’t cause issues…It would be helpful to see if there is any vascular compression, although this doesn’t always show depending on whether you get worse symptoms with your head in a certain position. Surgery is really the only ‘cure’, so in that sense, yes you would need to consider pushing for that.
It would be wise to to get your hyoid looked at in imaging, as we’ve had several members who have had clicking in their throat caused by elongated hyoid bone processes rather than the styloids, so technically hyoid bone syndrome. It can also be caused if the stylo-hyoid liament is calcified near the hyoid bone end (it’s more common to have calcifications at the top/ styloid end) , or it could be that yours is calcified the whole length.
Do you know of any doctors in your private healthcare group who treat ES? Are you able to see one of the doctors on the list? They both seem to be head & neck surgeons so good for both styloids & the hyoid bone if needed…otherwise I did find this research paper online:
saskia_boogaard_-_eagle_syndrome.pdf
I hope that you can get some treatment soon 
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