I went to the store and bought two locksets- a bedroom set to replace the one I took out of the backdoor (there is still a deadbolt, remember) and a new keyed entry set in case I couldn't figure out the old keyed entry set. I bought both knowing one or the other was going to go back to OSH (Orchard Supply Hardware) after I figured out which one I had to use.
Then I discovered (after trying unsuccessfully to shim it) that by screwing in the door knob the length of the lockset adjusted itself (hard to explain but it worked.) I got it in, got it adjusted, and in the process unintentionally disabled the pushbutton. It turns out that is a good thing, because it still locks (if you turn the button with key or finger) but the dogs won't be able to lock me out by jumping against the door. The old backdoor latches better than it did with the professionally installed lockset before, and the trailer door latches better and turns more easily, and best, works with the same key as the house.
Jack sat slept on the bed of the trailer while I did the lock and didn't want to leave when I was finished.
Successful weekend, except (so far) for the microwave
and the sneezing. (And the fact that the housework isn't done.)