G.B.
He is really too big to go to a toddler bed, the reason I say this is he is going to outgrow that mattress very quickly. I would put the baby bed mattress on the floor and set up a regular bed. He is going to be transitioning to a toddler bed and by summer or fall he will be transitioning again because he won't be sleeping on his bed anymore and no one will know. He will have become too heavy, large, for it and the inner springs will be uncomfortable, his feet and arms will be falling off and waking him up, he will not be able to find a comfortable position on it.
Toddlers are 12-24 months old. Once the hit that 24 month age they are classified developmentally as pre-schoolers. The same is for the toddler bed. He may be too big for it very quickly.