Just my two cents:
a) True connecting tickets are often (much) more expensive, particularly if you managed to get a special to LHR, but HAM was not on sale.
b) Interlining bags through LHR is not a preferred option if your luggage is dear to you. Yes, you would receive it eventually, but on one route (with a 3 hr connection) I have not received my bags more than half the time.
c) As was pointed out, if you have seperate tickets, interlining can be done, but it is at the discretion of the airline. So donīt be mad if they donīt do it.
d) Where I see the only problem, is that you are AA PLT - OW Sapphire. They should go an extra mile to make you happy. However, I would be really surprised if they still wouldnīt do it for an OW Emerald.