The PIC can claim the drawback individually irrespective of who owns the plane
Actually anybody whatsoever can submit the duty drawback form to C&E, which is why anybody in a "multi pilot" arrangement needs to be careful to check that nobody else has already done it before.
This is a not unknown scam: a renter puts in the drawback request, pockets the refund, and the aircraft owner then can't do the reclaim. When I used to rent mine out, one "IR instructor" did this to me, and since he departed with full tanks and only flew across the water to LTQ, he got free fuel for the whole flight. Another instructor did the same a bit later. One has to make sure, because if a duplicate claim goes in, and they catch you, they won't like it.