It sounds like it's on the load side of the gfci "gfci protected" this could still be a light issue or possibly the gfci itself, or the way the duplex receptacle is wired or the condition of the wire.
Easiest thing first. Try it in another gfci to see if it trips in the same fashion, if not it points more to an issue with this tripping gfci or the receptacle it is protecting, but it could still be a light issue, if it doesn't trip the second gfci you try it in, do what you are most comfortable with, that may be trying to get a light replacement, but if a second known good light has the same issue I would look into getting a gfci checker and see if it says open ground or reversed polarity, if the checker checks out i would look into replacing the tripping gfci.
There's a few easy steps to try before messing with the wiring.