Is there such a thing as control logic complexity classes?
One should be careful about having a head that large. If your tape head can read the entire input, then technically speaking all inputs can be decided in one step. This is because "making a head so large you can read the input" is formally equivalent to making the tape alphabet so large that it contains all input strings. But if this is the case then your entire function can be encoded in the transition table of the Turing machine, so every input can be decided in one step. It makes perfect sense to have a superconstant number of symbols in the TM, but when you do that you are no longer talking about "uniform" computation (fixed-size programs) but rather "non-uniform" computation, where the size of the program can grow with the input length. In complexity theory, non-uniform computation is typically studied in the language of circuit complexity: having a polynomial-size Boolean circuit family for solving a problem is equivalent to having a "polynomial-size transition table" for solving a problem with a polynomial-time TM, and the general feeling is that circuits are more natural to reason about. But it's possible that the TM perspective could be useful too.
Allowing a specific number of states or specific structure probably will not change the definitions of the standard complexity classes. This is because there are universal Turing machines which have a small number of states and/or simple structures in the transition table, which can simulate arbitrary Turing machines with polynomial time overhead. (Sorry, don't have references at the moment.) Restricting yourself to these machines does not change the overall difficulty, up to polynomials in the running time.