Yes, as others have posted, D and E are connected by an airside hallway, so one can connect between AS and AA without clearing security. Unfortunately, this does not help a BR member with lounge access.

In my experience (using a priority pass lounge), I was given access to F terminal to access the lounge there, with a same day boarding pass out of D; the TSA agent seemed to indicate this was commonplace. Clearly YMMV.

As for your return connection. It would be hardly any more painful to connect out of FLL than MIA on an international inbound to MIA That is to say, MIA is pretty much the most painful US arrival airport ever, and getting back to your connection is a nightmare. Last week though I saw signs indicating their forthcoming improved facility...here's to hoping.