Can peer-to-peer payment platforms transfer money across state lines without obtaining money-transfer licenses?

No. Moving money across state lines and/or internationally (across borders) with respect to the US, requires a Money Transmitter license.

We had a client that required this clarification, and we got this not only from FinCEN, but from a local law firm as well and both say the same. If you're moving money (physical, digital or alternative), other than being an MSB, a Money Transmitter license is required.

Facebook, on the other hand, forbids its platform/canvas for being used for any sort of payment system other than Facebook credits. P2P payments are not allowed on Facebook.

Just updating my answer: Facebook does have the necessary money transmitter licenses.