If you're the god of a species that dominates the planet you live on, why scrimp on space?
- Since the Doctor could sit in the chair, there must have been legroom inside it.
- Since Davros apparently has no legs, the cables poking out of his lower body must have been attached to some contacts inside the chair.
Ergo, the inside of the chair had more room than was necessary. Simple explanation, and the only thing that makes sense.
Alternatively, how about an in-universe explanation inspired by an out-of-universe one? Perhaps there wasn't really specially designed legroom inside the chair, but the Doctor could crouch with his legs folded beneath him. Just as Peter Capaldi did in reality (image links to Youtube video):