I tried adding one prop at a time instead of two. It did not matter.

I re-imported the combined prop with no options checked, including weld identical vertices. No difference. 

Note that I can combine these props with a different, less complex prop without problems. When I try to combine them with a different, equally complex prop, the same vanishing-polys thing happens. When I say complex, I mean less than 50,000 vertices, which is really not all that complex.

I'm starting to think the only solution might be to do it as a figure, with all new added pieces being body parts. I don't really like that solution because I try to have as few figures in my scenes as possible. More than a couple figures and Poser becomes slow and jerky.