Is there a way to add gold to it? Whenever I go to the door it just tells me how much I have, but not with any option to withdraw or deposit. Just want to make sure that isn't something that should be happening, and if it shouldn't be, then can I ask why adding to the stash with earned gold isn't something we can do?

I just got to the cave and built the inn. Judging from what context clues were given it most likely isn't interactable and is just to explain  how you could pay to build your fortress. Granted I could be way off but from the dialog thus far that's what it seems to be.