So, fortunately, this isn't a worry. The Pathfinder Core Rulebook FAQ includes this exchange:
Do ability modifiers from the same ability stack? For instance, can you add the same ability bonus on the same roll twice using two different effects that each add that same ability modifier?
No. An ability bonus, such as "Strength bonus", is considered to be the same source for the purpose of bonuses from the same source not stacking. However, you can still add, for instance “a deflection bonus equal to your Charisma modifier” and your Charisma modifier. For this purpose, however, the paladin's untyped "bonus equal to her Charisma bonus (if any) on all saving throws" from divine grace is considered to be the same as "Charisma bonus (if any)", and the same would be true for any other untyped "bonus equal to her [ability score] bonus" constructions.
Thus a composite bow with a +1 or higher Strength rating gains its damage bonus from the wielder's Strength bonus, and a composite bow with the weapon special ability adaptive likewise gains its damage bonus from the wielder's Strength bonus. The game says those are the same source—the wielder's Strength—and don't stack.
From the special ability's publication in Ultimate Equipment until that Oct. 2014 FAQ entry, it seems like it may have been possible to make an argument that the strength rating damage and the adaptive damage stack—such an argument likely hinging on adaptive's errant as if, a phrase inevitably leading to unresolvable disagreements. My research yielded no one during that time having actually made that argument, and this GM would have ruled against it had the issue been raised at his table.