Sure why not. But i think you need to define that when something is large, what characteristics does it have. For example, does the Large trait carry an additional +2 HP per level and does an additional damage base damage die to accommodate its size, maybe a +d4 due to it's size or maybe just a +2 (do what you think works, and adjust). Then on a larger weapon, how do you want to handle its base damage, double, triple, etc? There have been some good comments on weapon size here.
As to a Large creature being given disadvantages, I wouldnt do that. What i mean is, i wouldn't do that just for the sake of doing it or to balance anything. This is life, if you have the Large trait and can't fit into a tunnel, so be it. The PC just needs to deal it. If people distrust you because you're Large, it should be because of experiences with the race you are or from a past experience. Not simply because you have the Large trait. I grant larger things are easier to hit with a missile weapon just as smaller things are harder to hit. So take that into account, but this should already be taken into account anytime you're firing missile weapons, just a function of combat. And i add, being Large shouldn't be a class, or special feat, or special to an NPC, monster, PC, etc. If you're Large, then you're Large period. Give that a try and see how it plays. Then adjust as you go... and have fun.