If I found an unconcious diver on a recreational dive. I would not mess around with trying to find a weightbelt or weight pockets. I would simply grab them and inflate their BC, and hang on tightly. I am pretty sure I could modulate the amount of air in the victim's bc (as well as my own) on the ascent.
If I pressed the up button and no air came out or I saw that air was just venting from their BC, then I would mash MY up button, and hang on. I generally have a decent amount of excess lift capacity in my BC and this exact scenario is one reason why I don't try to minimize the lift capacity of the BC's I buy.
Dropping lead would be my third option for bringing a diver up, partly because it is more complicated than pressing a button.
From a strategy perspective, it makes sense to try to first inflate the victims BC, because if you should accidentally "drop" them, or loose contact (for whatever reason), once you made them buoyant, you can be reasonably sure they will end up on the surface and not sink back down to the bottom. Also, operating the victim's bc is pretty easy, since you can readily see the device, it may even be easier than grabbing your own.
The challenge is to also vent your bc on the ascent, you will need to manage two BC's; if their BC has the capacity, it might beneficial to completely vent your BC on the bottom, so you have only one container to deal with, but that is assuming a lot of moving parts in a life or death emergency.