What notable automaton models have polynomially-decidable containment?
Visibly pushdown automata (or nested word automata, if you prefer working with nested words instead of finite words) extend the expressive power of deterministic finite automata: the class of regular languages is strictly contained within the class of visibly pushdown languages. For deterministic visibly pushdown automata, the language inclusion problem can be solved in polynomial time. For more details, see the paper by Alur and Madhusudan, especially Chapter 6.

By the way, the nondeterministic variant of visibly pushdown automata is exponentially more succinct than the deterministic variant, but there the language inclusion problem is EXPTIME-complete and thus intractable. 

Alur, R.; Madhusudan, P. (2009). "Adding nesting structure to words". Journal of the ACM 56(3): 1–43.