As a playable character in a A:I-styled AVP game, I think the Predator should have a limited arsenal. No Plasma Caster, at least not until the very end, perhaps, no combistick, not even an Alien vision mode. It's only long range weapons would be the shurikens.
The Predator campaign could take place in one of the Predators' pyramids, where one Alien survivor from a previous hunt resides. The Predator in question isn't even on a rite of passage. It only wants to take on the Alien, with a very limited arsenal, and claim it as a trophy.
There wouldn't be only one Alien though, but also a chained Queen, and the eggs "she" lays. There would be many facehuggers crawling around.
Alternately, the story could could be about a Predator captured by the BSPs, that is forced to fight for it's life against an Alien, again, with a very limited arsenal, and no helmet. The captured Predator could be either a Classic or a Black one, it doesn't matter much.