I don't think so.

Let us assume that Reality is indeed deterministic, and that this inevitably and certainly leads to the impeccable logical inference of the non-existence of free will, of choice, of critical thinking etc.

OK, only one thing remains to be asked. Why would logical reasoning be a good method of dealing with the problem?

The only answer can be: because logical reasoning, it's election as a method, its development and its inevitable conclusions -- the full package -- is also, inevitably given and pre-determined.
If reality is deterministic, we are deterministically driven to use logic to prove that reality is deterministic.
And there you are, stuck in a circular reasoning, therefore invalid according to the canons of logic itself.