Shelf-stable "frosting" for a savory "gingerbread" house?

Peanut butter might work. It is safe at room temperature and not really sweet. It is somewhat soft, but so is the canned cheese. Natural peanut butter with the oil poured off might be thicker for use as mortar. The flavor would be good with pretzels.

Maybe, just maybe, meringue will work. You'll have to dehydrate it after assembly, putting it in the oven at low and slow, like a baiser.

If you want to give it a shot, try it out first. I'm not certain it will hold well enough on the slippery pretzel surface, and it's also possible that the logs will become unpleasantly dry.

Another option would be to try something based on a vegetable paste, like ajvar or kyopolu. But while the paste itself is preserved, it is wet enough to wet the logs through so they could be unsafe at room temperature and/or get an unapetizing look. Maybe binding it with gelatine or similar will work.

Possibly you could experiment with something fat based, like buttercream without the butter. Solid fats can actually whip well without sugar. It will be rather heavy though, so maybe emulsify one of said preserves into it.

Personally, I'd also welcome a whipped ganache made from 99% chocolate and cream, without any sugar. But this is unlikely to appeal to children.

Finally, molecular gastronomy may offer a way out. You could try to make a set foam from something which does not have enough nutrients to spoil, in the simplest case just colored water. Again, you have to work in a way which keeps the logs from getting too soggy.