As OP, I appreciate the great comments and advice.

I did look into using Avios on AA from AUS to JFK, but there was no award availability, and the lowest revenue fare available was much lower on B6. Luckily, we still get a free checked bag each on B6 as this was purchased before they changed their free bag policy.

My Vueling flight was booked with Avios on Iberia.com. I sort of assumed that since BA, IB, and VY were owned by the same holding corporation, they were also partners. My bad. Even though BA and IB are partners, and IB and VY are partners, that does not imply BA and VY are partners. The transitive property does not apply to airline partnerships.

I am fully prepared for the worst case on the outbound, where I will be collecting bags in JFK, and then again in ORY. It's not the end of the world.

On the return, we are taking the TGV from BCN to Paris, so it'll only be ORY-JFK on BA and JFK-AUS on B6. Since I have to claim bags in JFK for customs anyway, the only question is whether the recheck area outside of customs will have a B6 desk to check my bags in. If not, we may have to shlep over to T5, unless I can get the BA checkin agent at ORY to tag the bags through to AUS.

I do think at this point, we should wait for the empirical evidence. I will report back on this thread in mid-Aug after my trip!