You definitely need to give him more room regardless of the clean up. The cage is small so having him in there will cause problems to develop because he cannot move freely. This can escalate into sanitary issues, sore hocks from inactivity, obesity, and destructive tendencies for not having freedom.
I suggest using the playpen set up for training sessions where you do pick him up and put him on the box at every accident, poo or pee. Designate a sound so the connection is more clear. For Wick, a snake hiss sound very much spooks him, and that’s what I use. If I’m across the room and I see him perching to pee, I make the hiss sound and he immediately dashes to his litter box. This is the kind of connection you want made with the litter box training.
The answers provided in this discussion are for general guideline purposes only. The information is not intended to diagnose or treat your pet. Seek the advice of your veterinarian or a qualified behaviorist.