I haven't had a lidocaine infusion, but I have read about them, and they do seem to be mainly used as a diagnostic tool. I don't think any possible benefits will be long-term...
The main red flag for me is the rush/haste/slapdash/can't-be-******d attitude of your doc. There are significant risks invoked in the procedure, and you as their patient should be completely informed as to WHY they think you would benefit from doing this, and most importantly WHAT THOSE RISKS ARE. It is completely negligent for any doctor to try to sleep-walk their patient into doing a procedure.
It happened to me with my knee

and is the reason I now have CRPS in my entire left side, and my right foot. I believed every word the surgeon said when after physio wasn't improving my pain, he declared he would just have a quick look around in my left knee and "fix whatever's wrong in there". I didn't ask enough questions, didn't research the procedure and the risks enough... I know I was just darned unlucky to get CRPS from it, but....
My point is, ESPECIALLY if you already have CRPS, do not go into anything without all the information you need. Of course they are pushed for time, and as a patient you do have to research lots aout it yourself, but any remaining questions and concerns - if your doctor won't make the time to talk you through them, then don't let them do it. Our lives and bodies and futures are far far too important to put them at risk without a darned good reason.
Sorry to rant. But your whole post really flagged to me that you are confused and scared and a bit lost - and it's your doc's responsibility to help with that, and to do their best for you. You deserve no less than that.
My left foot is now my worst affected place, and I was in denial for as long as I could too

. It's horrible to accept that this can I just take over some other part of you... Keep it moving and flexing, massage it, do the warm epsom baths, and use it use it as much as you can.
Good luck with everything, I hope (whatever you decide) that the future is brighter than it probably seems right now!
Bram.