Without some better definition of "fact" and "theory" the question makes little sense to me, and hardly seems resolvable by simple logic.
It looks more like an example of the self-refuting nature of skepticism. Or an antinomy based on a false distinction. If we are to take (3) as a "fact" of some sort, then isn't it refuted if (1+2) constitutes a rudimentary "theory"... and equally refuted if it does not?
Or it is refuted by something like "common sense" in the way that Berkeley is. There is no convincing basis for "theorizing" that there are "no theory-independent facts," just as we cannot "prove" there are no "mind-independent objects" without recourse to mind, and it all tumbles into nonsense.
On the other hand, it is easy to accept some pragmatic idea of "fact" and that there are no absolutely stable facts because there are no absolutely completed theories. Nothing wrong with that. The "for-now-sort-of facts" are fragilely embedded in some Quinean web.
In any case, it seems to me there is nothing analytically wrong with (1) and (2), provided (1) gets rid of the vague "we cannot" to read: "No facts can be established without a theory." One could then assert, provided "facts" are things that must be "established," that there are no "theory-independent facts." Pretty meaningless.
But (3) as given is just a non sequitur. It requires something like (2.5) "There are no theories" or "There are no theories fully established by the facts." Then the circularity becomes evident.