What I did was to stay in both hotels; my trip was somewhat the reverse of yours as I was going from San Paulo to the Falls to Buenos Aires. I was very glad I switched hotels as it's nice to see the falls from both sides. Both hotels are just a short ride (IIRC at a high fixed price for the distance) from the respective airports. For your trip, you would presumably fly domestic from AEP to the Argentina airport at the falls, take a taxi from the airport to the Sheraton, stay there about two nights, move to the Belmond (crossing the border in the process, but it would be best to have the Belmond send their car service for you so you don't have to transfer to the hotel shuttle at thepark entrance--stress to the driver that you have a reservation at the hotel so you're not sent to the tourist buses and the regular tourist entrance and museum), stay at the Belmond about two nights, and then fly domestic from the Brazil airport to Rio, probably into the domestic airport at Rio which is much closer to downtown than GIG and about the same distance from Barra as GIG. [I don't know where Awasi is located.]

Both Copacabana Place and the GH had good security on hotel grounds. You don't need private security or to worry about being high profile especially as Rio has pretty extreme income inequality with very wealthy people and very poor people, so there are lots of "seemingly rich" victims, with muggers being interested in iPhones, passports, costume jewelry, and small amounts of cash. Just follow the precautions of not wearing jewelry outside of the hotel, taking only the credit card you need and small amounts of cash in a secure front pocket or better, don't take anything to the beach, etc. Rio would be a good place to use the hotel car service to get around, although this could get expensive. I felt safe walking around the neighborhood of the GH, but there have been reports of problems on the "executive" = air conditioned buses from Barra to downtown.