The argument would be rather complicated, with several sub-arguments, but the basic steps would be something like this:
1) A sub-argument that there is a natural inference from the way our choices seem to us (i.e., that they seem to be choices from among real alternative possibilities) to rejection of determinism in the case of choice. By 'natural inference' I mean an inference that is reasonable (although it could conceivably be proven wrong).
2) A sub-argument that we cannot make sense of choice without appeal to real alternative possibilities.
3) An argument based on (2) that acceptance of determinism requires holding that there is a fundamental and irresolvable contradiction in the way the world seems to us (between the perspective of choice and the deterministic perspective).
4) An argument based on (3) that the determinist has to privilege, without any good reason to do so, one perspective (sort of evidence) over another, even to argue for determinism at all.
5) An argument that the defender of the view that we have free choice is not committed to the irresolvable contradiction in (3) and therefore does not have to engage in the arbitrary privileging of one kind of evidence over another that is noted in (4).
6) A sub-argument that, even on its own terms, determinism is inadequately grounded in evidence given the strength of its claims, because 1) given the evidence of the way our choices seem to us, the empirical evidence required far exceeds anything that has ever been offered; 2) which has indirect confirmation in the fact that the arguments determinists historically have used to persuade people to their position are sophistical (this would actually be very easy); 3) and which is the most likely conclusion given certain sorts of infinite-regress-or-else-no-explanation problems that arise on deterministic suppositions.
A similar line of thought could be used to defend freedom of the intellect as well as freedom of the will, i.e., the fact that we are free in reasoning, deliberating, and deciding as well as in choosing.