He's barely old enough to stand up and have any power in his arms. My first thought is that his mattress is too high. Put it down all the way.
If it's all the way down then he must be really tall! Our baby bed had a lot of space above the mattress.
Other than that I guess you can put him in a pack and play. Maybe they have more height.
I do think he'll be fine to move into a toddler bed when he's 14-18 months old though. In child care we start teaching the infants to sleep on a cot around 11 months so when they move up to the toddler room they will be used to the cots. Not so much change all at once.
So I moved the kids onto the toddler bed as soon as I could after they moved up to the toddler room.
It worked great. I had a gate across the door of their bedroom. I had doorknob protectors on every single bathroom door, exit from the house, any extra rooms, and on the closets too. They keep them out of places they don't need to be. Bathrooms are the most dangerous room they can get in. They can drown in the toilet or tub.
All the small shelves were fastened to the walls with L-brackets and there weren't any drawers or anything like that because we put a shelf in their closet with their socks, jammies, and other folding stuff in them. They were about a foot long, 9-10 inches across and about 10-12 inches deep. I labeled them and it was easy to keep them organized. I hung all their clothes.
If they got up in the middle of the night they were safe, absolutely NO overage toys in their rooms. Ever. Nightlight in the hall and a baby monitor on a shelf in the hall where I could hear everything going on in their rooms.
It can be done. He's too young for this now and I think his bed needs adjusting but maybe he's just tall.