I couldn't say for certain, and every insurance company is different, but I really doubt they would deny you treatment based on the spinal tap (LP). From what I was told, the LP is not indicative of MS in about 20% of cases, so if it's negative that doesn't rule MS out. In fact, many places don't even use it, and rely mainly on MRI and clinical signs (I didn't have one, because I was told it wasn't needed for diagnosis). Also, usually if the doctor gives a diagnosis, that's all the insurance company requires to cover medication.
So, if you get the spinal tap to try and find extra evidence for MS, in my opinion, I wouldn't worry about it affecting the insurance.
I would be careful to ask the neuro about putting a label on what type of MS you have (i.e. progressive or RRMS). Because, if you do want to give a DMD a try, the insurance company may not cover a DMD if your are diagnosed progressive. But, I do not know this for sure.
Also, I'd be a little wary with the lyme docs.