I can imagine the plausibility of an Avis dynasty in exile in the Açores, but that would probably entail a militarization of the populace and society of the archipelago to face multiple foreign threats, support from other powers, a stable source of income, even if piracy (which would kind of weaken Antonio's claim to the portuguese throne), a sailing tradition in the isles so that fielding a powerful navy is plausible, the fortification of key centers to face the (inevitable) spanish invasions to come, and maybe the development of technical skill inherent to the aforementioned topics that leads to a modrnization of the fleet, maybe smaller, more maneuvreable ships with good firepower, archipelago frigates and what-not.
On the other hand, I can also see the possibility of an exile in the colonies, a prosaic resource very common in ATLs. In this case, he would probably go to India, the most secured part of the empire at the time, and just to have a notion of the state of the empire, the Indic possessions of the Portuguese crown were constantly endangered by local rulers and foreign powers. Maybe another option would be Brazil, but the French and Dutch had their eyes on time for quite a while and would probalby take the opportunity to seize the portuguese south american possessions almost like OTL f the spanish were more distracted, so a pseudo-butterfly of such a timeline would be a completely different Brazil.
Furthermore, taking the second option into account, if Antonio and maybe some loyalists went into exile to the Indic, probably with French and English ships and supplies, he would have a tough nut to crack, facing simultaneously foreign plots, native rebellions and gubernatorial treachery. It would take a lot of time and energy to solidify his hold over such territory, and developing the means to secure it all and maybe even expnd it would be very difficult indeed, especially if we have the threat of spanish intervention and loyalisms in mind. And one must make it so that the probably-sick-of-that-game French and English wouldnt just seize the Portuguese outposts around the world and tell Antonio to f--- himself, leaving him without support, a base of operations, money and men.
Another option would be that, after the seizure of Portuguese colonies all over the world by foreign powers and native rules, Antonio would turn into a pirate with a few loyalists and some quick ships, seize some forgotten but well-positioned island, probably in the Caribbean, and turn to piracy to fight foreign interests in his now super-reduced "empire".