Is there any polynomial time algorithm for a restricted EXP-time problem?
Your first formulation isn't very useful, as either the machine which simply writes $0$ on its tape (without putting in any effort) or the one which writes $1$, satisfy your condition. Fixing both $T,w$ means you have only one question at hand, and not a computational task we can reason about in terms of the amount of resources require to handle it. In that case, a simple yes or no consists as a valid answer, and we can output it in constant time.

In your second formulation, you're asking whether the language

$L=\{(\langle M \rangle,w)| \text{$M$ halts on $w$ within $2^{|w|} steps$}\}$

is decidable in polynomial time? The answer is no, since $L$ is hard (under polynomial time reductions) for the class $\mathsf{DTIME\left(2^n\right)}$. This means that if $L\in P$ then $\mathsf{P=DTIME\left(2^n\right)}$, contradicting the time hierarchy theorem.