Complexity Classes for Cases Other Than "Worst Case"
Although I can't quite parse your definitions, you should know that much stronger time hierarchies are known, in particular "almost everywhere" time hierarchies. 

Here is the formal statement: for every time bound $T(n)$, there is a language $L \in TIME[T(n)]$ with the property that every deterministic algorithm that correctly recognizes $L$ must run in time asymptotically greater than $t(n)$ on all but finitely many inputs, for sufficiently small time $t(n)$.   

"Sufficiently small" means $t(n) \log t(n) \leq o(T(n))$.

Suppose we choose $T(n)=2^n$ for an example, and obtain a hard language $L$. Then, any algorithm that correctly recognizes $L$ must take at least $2^n/n^2$ time on all inputs past a certain length. This seems to be what you're looking for in your class BestE.

Reference:


  John G. Geske, Dung T. Huynh, Joel I. Seiferas: A Note on Almost-Everywhere-Complex Sets and Separating Deterministic-Time-Complexity Classes Inf. Comput. 92(1): 97-104 (1991)