I think the best way to insert Makar and Measse into the later Legendarium is to make them Maiar of Tulkas, who came with him to Arda to help the other Ainur fight Melkor, but later turned out to be rebels. Either they joined Melkor for some time and then left him as well, or they realized early on that they wanted to live separately from both Melkor and the other Valar, because neither side could offer something that would fully satisfy them. Melkor was too destructive and too oppressive as a master for their liking, and the Valar were too boring and also restrictive in their own way.
That’s how they could end up living in their own palace, with a bunch of other Maiar hanging around, and sometimes causing some shit in the world, but sometimes helping to fight Melkor (like they did in BoLT).