My dad and I bled the brakes tonight. My dad noted that he could pump up the pedal against the air bubbles, which it wouldn't really do before. It also wasn't making nasty popping noises like it was before. It still made a little groaning noise from the check valve each time the pedal hit the floor, but the valve I got with the new booster makes the same noise if you blow through it just right, so I suspect they just do that, same as the rollover valve on the evap line from the fuel tank. We went in Nissan's bleed order, but also bled by cracking the fittings at the master, and got a little air out of each one. The pedal felt good after that but the rear brakes still weren't doing much. They'd stop the wheels, but I could push the tire with my foot and turn the wheel. My dad suggested we take it out, romp it, then try bleeding it again in case the rear circuit still had air in it. We took it out, did a few panic stops, and managed to lock the front wheels on dry pavement, which I'd never gotten this truck to do before. The next bleed got a bunch of air out of the driver's rear caliper. We tested it again with the wheels in the air and hallefrickinluyah--the rear brakes are back!
I tore apart the old booster (straighten the waves in the ring around the back and it all comes apart without having to cut anything) and found that it was surprisingly clean and nice inside. The rubber diaphragms (it has two, which I wasn't expecting) looked brand new. The play in the output rod is supposed to be like that, near as I can tell; nothing on the output side was obviously damaged. Given the different part #s I suspect the one I put in is just a slightly different design. Some of the boosters at the wreckers had that play and some didn't. The one glaring issue I found was the internal air valve that makes the booster work. It was dirty (the only dirty part in the whole booster) and it caught and popped in its bore when I worked it back and forth. When you press the pedal, the input rod compresses a spring, pushing this valve spool into the plastic chunk it lives in, which allows air from the cab to fill the rear chamber of the booster, pushing the diaphragms into the front chamber (which is under vacuum). The plastic chunk pushes forward against the output rod, both from the force applied to the pedal and from the force applied to the diaphragms.
I'm not 100% sure how this cluster unfolded but my best guess at this point is that the booster inhaled something it shouldn't have, which I pushed deeper into the valve while bleeding the brakes, which made the valve bind and catch, which limited the effect of the booster, which limited brake pressure, which wasn't high enough to activate the prop valve, so the rear end didn't get enough pressure to do much of anything... and what I thought was the master bottoming was actually the the valve catching in the plastic and trying to push the brakes directly without vacuum to back it up. TL;DR the booster was hooped and my brakes are back! Thanks to everyone who followed this clown show and threw suggestions at me. Hopefully this thread saves someone else from spending two months pouring brake fluid on the floor.