What are acceptable treatments for chronic pain that will not result in a loss of an FAA Medical?
The only real answer here is to consult an AME. I searched on the AOPA forums (members only) and the information wasn't encouraging: all current (as of 2014, which was the more recent relevant post I found) neuropathy drugs are disqualifying. AOPA's list of approved drugs does show some approved drugs for pain relief, but it looks like it's all OTC stuff.

But, the usual medical disclaimer applies here: every case is different so the above may be irrelevant to you personally; I'm not a doctor and I don't know what I'm talking about anyway; only an AME can give you meaningful information.