I am afraid you wasted money taking him to a vet for this non issue, and the vet is a quack, now that we have that out of the way here is the root of your problem (see part in bold)
I have a 5 month oldYou are underfeeding your BP.I feed him pinkies
Pinkie rats or mice are WAY TOO SMALL for a BP that age.
An hatchling out of the egg will start on hopper mice or rat crawlers.
Hatchlings and juvenile utilize most of their prey for their growth, living little to be defecated, now if you are underfeeding your BP this means NO poop.
So the issue here is that food = poop, what goes in must go out, if nothing goes in or not enough than the little that goes in is being used entirely.
One has NOTHING to do with the other, you do not need a weight to switch to a bigger prey, you just need to feed a prey equal the animal's girth size.I wanna move him up but until I get him to poop, I can't get an accurate weight and see what size I should move him up to
Now as far as empty weight your BP is empty anyway see above.