Avios are only redeemable on AA only when AA is offering "MileSAAver" awards on a particular flight. (And BA charges separately for each flight flown, so if you have to make a connection on AA, you'll be charged more than if you were on a nonstop.)

AA miles are redeemable on JetBlue, but only on flights where JetBlue is making award seats available to partners.

I don't know what they cost, but there appear to be frequent flights between AUA and CUR. If they are cheap, I would look at getting an award redemption to AUA, and then a cash ticket for the inter-island segment. JetBlue flies JFK-AUA nonstop, and AA offers seasonal LGA-AUA nonstops on Saturdays only.