This is a common predicament because a rabbit’s sleep and wake cycle does not line up perfectly with a human being that likes to wake up after 5am.
While it’s great you have him out so often, being confined is a deterring feeling for him. It’s like letting a prisoner be free for the week, but put them in a jail cell on the weekend — the weekend is when you want to do things, regardless that you were free from prison for five days already.
You can try and keep his cage now stimulating at night by around fresh hay and saving pellet time until night so he can munch and sleep. Attaching an xpen outside the cage at night could give him some more room so he feels more satisfied and less confined. The xpen could just be set up at night so it’s not a permanent, large space hogger in your room.
I made Wick a larger cage, but getting him quiet was too difficult honestly. It’s not something you can punish for because it’s literally just how a rabbit’s body functions. Wick is just free roam 24/7 now and he’s my alarm at 515am, which works when I need to get up for work. On the weekend, I just pet him enough that he falls back asleep with me so I can squeeze in another hour or two. Or he realizes I’m not getting up and goes to eat hay.
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.