It's been a while, but I have an update. After doing a number of engine flushes/treatments/etc. I took the van into another shop to get the oil pan pulled off and pump replaced. Before doing that, however, they put a pressure gauge on and drove the van around for a while. It seems that pressure from the pump is just fine, even when the lifter problem occurs. They also stuck a camera down there and didn't see any crud or particles down in the pan. So that eliminates the possibility of the oil screen getting clogged or the oil pump being bad.

The mechanic said the remaining possibilities are either bad lifters or a clogged line in the engine block somewhere, which would be very difficult and expensive to locate and fix. They quoted me $650 to do the lifters, so I'm just gonna do them myself. I'm a relatively inexperienced mechanic, but my understanding is that this is not a terribly complicated job, correct?