You may want to consider an upgrade to Club Europe to get the third bag checked through. You don't have to recheck them in Paris, so long as the BA agent in LHR can see the relevant details (flight number, locator, name on the AA booking etc), I often suggest putting a post-it note on your passport.

If on a revenue ticket, bought from BA, then you may get a POUG - cheapish upgrade to Club Europe. If a redemption, you may want to consider changing the redemption.

For carry-on it's size rather than weight that matters, and whether we agree with it or not, there is sometimes a degree of latitude granted, particular if one or both of the potential carry on items does not have wheels. But you may get an agent who sticks to the rules (not many such agents at T5, but presumabaly they do exist!).