The thing about the Golden Company is that it's basically staffed by Westerosi exiles, some of them criminals who would have normally been sent to the wall, if they hadn't gotten on a boat fast enough. That said, some ideas on a hypothetical permanent Golden Company, to support a Blackfire-ruled Westeros even after it is set up:
-The Blackfire-backed claimant gets the option to either disband the Golden Company after taking the throne, or keeping them on staff; disbanding it should have some sort of fantastic bonus to it, such as giving the all the gold of the current head of the Golden Company has to the Blackfire clamant or giving the kings immediate holdings a very large temporary (20+ years or so+) levy recovery boost, to represent the various members of the Company resettling Westeros as free men. The levy boost should be tempting, because after the war that would result in the Blackfire claimant taking over everyone's army is likely heavily reduced in numbers. The Company could possibly be re-formed by any Blackfire adventurer against a non-Blackfire king, regardless.
- If a male descendent of Bittersteel is alive, or if Bittersteel himself is still kicking, he could get Driftmark as his just reward, for his services.
-The Company could be 'given' Kings Landing, and the lord of the Golden Company would get the Commander of the Gold Cloaks honorary title. Hence, the Company would always be around in Kings Landing and is able to raise itself at a moments notice.
-Some Varangian-style events, mostly with some 'dishonor' twists. Lords visiting Kings Landing getting a daughter preggers because of a 'disgraced' sellsword and such.
-The Golden Company should sometimes rebel against the current Blackfire heir or king isn't 'up to stuff'! The Company didn't fight a war to put up a unjust king, and a tyrant Blackfire should fear a especially happy cousin with friends in the Company! The Company is also fine with putting up any dragon with a claim; as long as the heir is legitimate, than they could put up a Targ if there's no suitable Blackfires.