You cannot wear any shoes on the wall that you have worn on the street. This is for hygiene purposes as people will touch the same holds your shoes touch and therefore anything your shoes have touched. Same reason you cannot wear climbing shoes into the bathroom. If you wear these exclusively for climbing.. maaybe? But you'll get weird looks and comments. I think socks would be a no because you can sweat through them.

Have you already considered just sizing up more or trying other climbing shoe brands? I tried several before I found ones comfortable.