Transfers between separate tickets are not "connections" and that causes significant confusion.
The simple answer here is that AA has no contractual obligation to rebook to anywhere, whether he is flying onwards with AA or any other carrier. Anecdotal evidence is that AA has spottily rebooked for internal policy reasons. But, OP has no recourse if it will not.
Thus, flying on separate tickets is solely a matter of risk tolerance. What will it cost to change OP's existing AA ticket to terminate in MAD vs. what will it cost for him to purchase a new LHR-MAD ticket on IB if he is a no show and has 4 hours less transfer time to no show.
Although OP does not say he has checked luggage, if he does, he will need to reclaim that at LHR and then drag it to IB to check it in. He must therefore allow time for passport control and customs (such as they are) and take into account IB's bag check deadline.