This trophy requires multiple playthroughs as do Joel's upgrades. There is no way around the fact that this trophy is incredibly time consuming. What I am doing is to minimize playthroughs is that I fully upgrade the weapons you get late in the game first. Such as the Flamethrower, El Diablo and the Shorty. And leaving weapons like the 9mm for last as those are easily upgraded.
NG+ is your friend when it comes to this trophy, since all your upgrades carry over.