If I understand correctly, Lyme is not often seen
in spinal fluid, and so a spinal
tap may not give the evidence sought. Since a spinal tap is not an insignificant thing to do, you might want to do a little research into that aspect before proceeding. (If the ID dr knows it's Lyme, why do the spinal tap, even if it were likely to be good evidence?)
There are other tests to do to see if the brain is affected (e.g.