This is totally outside my wheelhouse, so I could be totally incorrect. However, I think there (might?) be two different issues at play here. The first question (and pub 529) relates to whether the cost of a medical license can be deducted on Sched A as an unreimbursed business expense. That answer is clearly no for your first license, theoretically yes for renewal however it no longer exists at all, and used to be subject to the 2% floor which made it practically impossible for most to claim anyway.
But, my understanding from the question is whether you can deduct the cost of the license from moonlighting income. In that case, you're receiving a 1099 as an independent contractor. Assuming that you decide to file Sched C as a small business, then the question is whether that counts as a qualified business expense on Sched C, which (I think?) is covered by pub 535:
https://www.irs.gov/pub/irs-pdf/p535.pdf
I think (maybe?) that a medical license would qualify as a section 197 intangible asset (page 30 of the PDF) -- they mention as examples the cost of a liquor license or a taxi medallion. Seems very similar to me. Problem is, you need to amortize it over 15 years (which seems crazy, since the license is usually only good for 3-5 years, so seems you should amortize it over that period of time).
In any case, if you have to amortize it over a long period of time, it may simply not be worth it.
Any and all of this could be wrong. The only thing that I'm almost certain of is that if you're filing Sched C, get professional help. And by "professional help", I mean "not me".