I don’t think you have to wait for the REPL to be initialized — you should just do it immediately:
import REPL
REPL.REPLCompletions.latex_symbols["\\dflat"] = "𝄫"
It would be reasonable to add \doubleflat and \doublesharp (or maybe \flat2 and \sharp2, or …?) to the global table in a future version. Just submit a patch to REPL/src/latex_symbols.jl. We’re pretty permissive about adding new symbol tab completions.