Are there more polynomial time problems with complexity lower bounds?
Here's one involving a 2 player pebble game. You decide if it's natural (:

T. Kasai, A. Adachi, S. Iwata. Classes of pebble games and complete problems. 1979

Theorem 3.1 has the EXPTIME-completeness of pebbling. Theorem 3.3 has the easiness of k-pebble.

A. Adachi, S. Iwata, T. Kasai. Some Combinatorial Game Problems Require Omega(n^k) Time. 1984

Theorem 3.2 has the lower bound on k-pebble. Lastly, you might also be interested in:

T. Kasai and S. Iwata. Gradually intractable problems and nondeterminitstic log-space lower bounds. 1985

Sadly that these are all behind paywalls :(