Depends on who's asking for the average, but if you're bowling in a USBC-sanctioned tournament, the only average that matters is the one from the sanctioned league. As far as the USBC is concerned, that other league is just (organized) open bowling.
There could be a tournament out there that asks for both sanctioned and unsanctioned averages, but I haven't heard of one.
If your buddy at work asks what you're averaging, tell him whatever you want to. If your buddy is a lady with nice sweater kittens, I'd go with the more impressive, higher average. If your buddy is your boss and he's talking about a "friendly" game after work, go with the lower one and let him win.
SH