If it were me? I'd start with the rheumatologist.
My reasoning is: I don't know whether you've insurance or, if you, do: its protocols. For instance, to see a surgeon, my insurance "prefers" (for coverage) a referral be made from, i.e., primary care physician, rheumatologist, neurologist, etc.
If I were to see a surgeon, I'd select from a speciality you've not mentioned: A neurosurgeon.
My reasoning? Again, I promise I have some reasons
. The main one is due to what you've noted about your C-spine and pain possibly from your L-spine. I'd want a surgeon whose focus is also on nerves - possible compression, radiculopathy, etc. - whereas ortho. is more centered on bones.
Either a neurosurgeon or an ortho. could be good; I merely mentioned what type of doc. I'd be wanting to see next... having been there, done that (with rheumatologist, neurologist, neurosurgeon and ortho surgeon).