Decide if a string is in a language without simulating the automata accepting the languge
You ask two questions.

For your first question, about whether a TM can do this without fully simulating the DFA:

The problem statement is not well-defined.  Given a TM, you can't tell whether it "fully simulates the DFA on a string" or not.  (That's undecidable.)  There are all sorts of ways that a TM might effectively "simulate", but in an obfuscated way that makes it hard for you to realize that's what it is effectively doing.

Your language $L$ is certainly decidable.  It has the form $L=L_1 \cap L_2$ where $L_1$ and $L_2$ are both decidable, hence $L$ is decidable too.

For the second question, about whether a TM can do this using at most a constant number of steps after reading the last bit:

Sure.  That's always possible, for any decidable language.  After reading all but the last bit, the TM can decide what it will do if the last bit is 0 and what it will do if the last bit is 1 (doing as much computation as necessary to handle both cases).  Then, when it reads the last bit, it will immediately be able to produce the correct output, using only a constant number of steps.  This doesn't really have anything to do with whether the TM is or isn't simulating the DFA, though.