Have you tried frying the buns in the grease from just-fried burgers? Wouldn't work for lean ground beef, but the el cheapo hamburger, fried in an iron skillet, produces a traditional (pre-thick burgers, pre-franchise burgers) grease for frying the bun. Be sure to have add-on ingredients, mustard, pickles, whatever, at least at room temperature, not cold, before adding them.

If you don't want fried bread, at least toast the bread--maybe with butter and garlic? Putting a hot beef patty into a room-temperature or chilled bun is almost guaranteed to produce sogginess. Try to keep the temperatures between bun and beef patty at the same temperature before combining.

If nothing else works, maybe switching to stale, dry bread might work?

Anyhow, I am guessing that either temperature or moisture might be contributing to the problem.