This seems to be begging the question of how is it to be known that the God being worshiped by any particular Muslim, or Muslims in general, or any group whatsoever, is the "God of Abraham? Is it purely because of their claim that makes it so?
That would leave you in a rather dubious place.
Does that mean anyone who comes along and makes claims about how the “God of Abraham” is to be worshiped – converting non-believers by the sword, practicing polygamy, taking child brides, killing apostates, etc., etc., – is legitimately worshiping the God of Abraham?
Does the God of Abraham, for one, have any say at all, concerning what counts as legitimate worship of him? Is it left purely up to the determination of human worshipers to make these kinds of claims and, therefore, all such claims are to be taken at face value?
I remain skeptical that you can address these questions, while holding to the “cannot have it both ways” position.
Again, could you define why there are only two ways and not a spectrum of ways that define more clearly at which point worshipers are no longer properly worshiping the “God of Abraham?” Does it ever come down to how God is worshiped, or is that just irrelevant provided some kind of historical affiliation to Abraham can be shown. I would assume the theological details are irrelevant for you and that mere cultural or racial ties are sufficient.
Suppose, for example, a modern Muslim Arab, decides to begin a new Islamic religion with a whole new set of revelations, does this individual and subsequent followers get the bye because of their claim to acknowledge and worship the God of Abraham? At what point and using what criteria would you say, “this one has gone too far in how they worship God?” Or will you inevitably default to “You cannot have it both ways,” and always invoke the “God of Abraham” to render all such religious groups legitimate?