I strongly disagree with Phill.Zit's reading here. Your own quote says it all:
the weapon’s bonuses apply to the roll.
It goes on to clarify ('therefore…') this would require a finesse weapon. But bows use Dex as their bonus despite not being finesse weapons, so should use Dex for these checks.
In fact, I'll go further and say you have to use Dex. Finesse weapons give you the choice, whereas bows do not.
And note the wording on the types of combat maneuvers it applies to. Normally with a bull-rush or grapple
you're [not] actually using a weapon
so you'd be forced to use Str. But again, here you are using a bow so must use Dex.