He’s most likely getting out because he can see that there’s a whole new world outside of the bars. There are many inexpensive ways to cover the bottom of the pen, like getting a few big pieces of cardboard to block off the area (you could tie them to the pen).

The peeing and pooping everywhere is him marking his territory. Unfortunately bunnies like peeing on soft things, so it’s quite hard to prevent them from peeing on the bed. If you can catch him in the act, put him in his litter box as soon as you see his butt arch.

You’ll have to enclose him in a smaller area first, so the pen you mentioned may be a good start. Bunnies normally have a favourite spot to poop/pee, so monitor where that spot is within the pen and then put his litter box there.

If you see any pee outside the box, wipe it up with a paper towel and toss it into the box. Bunnies tend to pee in the same spot again if they smell their pee there, so get rid of the smell with a vinegar + water solution. Also toss his stray poops into the box. He should learn eventually that the box is where he’s supposed to go. Lots of bunnies love pooping and peeing as they eat, so you can put some hay in/near the box and he can munch on it while toileting.

Once he’s used to the box, he will be less likely to pee or poop outside of it and you can then enlarge the area he roams. You may still find occasional accidents here and there but it shouldn’t be too bad. If the area is too big, a second litter box may be required. Some bunnies don’t fancy running all the way back to their pen to toilet so an extra box would help.

What Kurottabun said – however my little ones were impossible to letter train until after they were Neutered as they would just mark their territory everywhere. He is definitely getting to that age if he’s 5 months.
Just a suggestion