My ash pan door latch, and main (fill) door latch, were operating a little roughly. I bought this lithium grease, which is supposed to work well in high temperature situations, at Pep Boys for $ 3.79. This is working well for the problem areas I mentioned, and the close-up of the latch on the fill door indicates that the grease is staying right where it should, and not melting away 16 hours after application. The darker areas of grease (on the peg, which should be the hottest of these parts since it's part of the body of the stove) might suggest it's drying out a bit. I'll watch to see if this progresses.
Any other suggestions for high-temp lube products people have found helpful?