I think it would be fun and not break it too much to allow the hero to take one non-class-specific item with them into the vault. That would take the start of the vault to be less deadly and make the run depend a bit less on the randomness in the room layout.
The catch is that the player would still be able to take just a single item out of the vault. So taking something into the vault likely means leaving it there in exchange for something better. And if not leaving it, it means taking nothing as a reward for the quest. Except for last shop, which is kind of a reward.
I think it fits the lore of the quest - the teleport to the vault magic is limited in the amount of things it can transfer between the vault and the regular dungeon. It just means the limitation is the same on the way in as it is on the way out.
I only succeeded in this quest about four times so far. And failed three times. It is quite deadly indeed. There was one run when I had good luck before that quest and had all the equipment I could've needed. That time I could've taken my +7 plate armor of thorns with me for a quick win and would've took it back out with me. Just to have the last shop unlocked.
And it does not require any new UI - the "select a single item to take with you" UI is done already for the teleport back.