modular tensor categories - What's the best reference for actual formulas for RT invariants?

Since no one else has jumped in, I'll try again.

Standard references producing a TQFT from a MTC include Turaev's big book and the shorter notes of Bakalov and Kirillov (available online here).

If you want something less algebraic and more combinatorial (which I doubt), I like the book by Kauffman and Lins, and also some paper(s) of Lickorish's from the early 1990's.

I haven't read the original Reshetikhin and Turaev papers in a long time, but I remember thinking at the time that they were perfectly readable.