How should I propose to poop?

Solution 1:

The Propose button actually is more of a "Recommend" button in-game. It basically proposes pets that you have that are more likely to synergize with the equipment that you have.

Remember that since certain equipment when worn together can give you special abilities, like Super Mach, Killer, Deathmarch, Paralyze, what Propose does is the game's way of recommending a different pet to add or change a different ability for your loadout.

Solution 2:

I'm 141st level and have seen three or four of these buildings by the 4.5 km mark. I have a big collection of weapons and pets. Sometimes when I hit propose it cycles through pets. Sometimes it cycles through weapons. My answer is, a specific bathroom will cycle through pets telling you their benefits, and others cycle through weapons.