A lot of people talk about policy, rules, etc. in this situation. I get it, AA has rules; however, the blind following of these rules is why AA gets bad press a lot of times. AA needs to give their employees flexibility to handle situations such as these before they get out of control either through a viral video, the news, etc. (refer to the OKC trip DL "saved").
IMO, the OP was reasonable here. He was travelling with an infant. He packed a carry-on to cover the needs of his infant for an overnight flight. You look at most airline baggage policies, and they make it clear to pack essentials in a carry-on. AA could not accommodate the bag as a carry-on to CLT, so the OP asked to pick it up in CLT so he could at least have it as a carry-on the overnight flight. Given the situation, I see that as a reasonable request. DL can seem to handle reasonable requests, why can't AA?
Also, I must be missing something on short-checking, because I have done this several times in the past on CO, DL and B6. If your connection falls out of certain time window, your bag can be short-checked and you can then re-check it in. The airlines have a bag-matching mandate so if he did not get on his flight to MAD, his bags would be offloaded.