No it's not unreasonable to ask for what is more comfortable for you.
I've had deep injections in both my neck and the lower spine. The neck injections made me very very nervous. The first place was going to just inject me in the office and I left. The second place did the injections in the hospital with the help of MIR to locate exactly where to place the needles...that helped a lot. With the lower back they didn't do that but they still used the hospital for giving the injections. The third place had their own machines but it was in a seperate area just for surgical procedures not just in the back area and they also worked with the fear the best. I spoke to a psychiatric nurse before hand who was with me during the injections. My cervical spine problems are better but I ended up getting a spinal fusion for the lower back. I put it off for years and now wish I had done it much sooner. I no longer need narcotics to deal with 24/7 pain.
Having injections in the spine means having a good relationship to the person doing the injections and being able to trust them . It's very reasonable to have these fears and to ask that they respect them and you.
Physical therapy, and acupuncture both helped with my neck problems and now I'm pain free. While I did it all( acupuncture{those needles did not bother me since they are very small} PT, chiropractic, diet etc) an old fashioned X-Ray from the side showed my spine was offset and I faced eventual paralyses if I didn't get it fixed. With the help of my T a really good pain specialist and a very skilled surgeon I'm pain free today. I need to keep up the exercises that's all.
So it's very possible to go though it all and be pain free. You can too, speak up and let them know what you need. I'm on Medicare and Medicaid so it is possible to find good docs out there.