When I was in a similar situation (AA schedule change that required rerouting, on a OWE issued on AA stock), the AA RTW desk was very flexible in giving me options to get where I'd planned to go. It sounds like you'd like to add a stopover in MIA, and, even if AA goes along with that, it's hard for me to see how you'd be able to avoid a reissue. I don't think there's any way they'd issue you a MIA-SJO boarding pass in LAX if you're doing a stopover in MIA -- that sounds like a reissue to me.

If I were you, I'd come up with a prioritized list of revised itineraries that you'd like (including the LAX-MIA-SJO-MIA-LAX solution that AA has proposed), and give the AA RTW desk a call. (Though I'd have to say that the prospect of a LAX-MIA-SJO-MIA-LAX mileage run in the delightful 757 gives me the shakes.)