It is true that you can use Löwenheim Skolem to get a countable model $M$ of ZFC assuming $Con(ZFC)$. But to use Mostowski you need additionally the well foundedness of that model, which doesn't have to be true, even though that model satisfies the axiom of regularity. The point here is that $M$ 'thinks' that it is wellfounded but from the 'outside' it is not.
Moreover, remember that $CON(ZFC)$ is merely an artihmetical statement, which doesn't tell you anything about the 'real' consistency of $ZFC$. So assuming $\lnot Con (ZFC)$ will not prove you anything you want, it will just prove you that 'there exists a proof for anything you want' (the statement 'there exists a proof...' is again an arithmetical statement).
It is true that you can use Löwenheim Skolem to get a countable model $M$ of ZFC assuming $Con(ZFC)$. But to use Mostowski you need additionally the well foundedness of that model, which doesn't have to be true, even though that model satisfies the axiom of regularity. The point here is that $M$ 'thinks' that it is wellfounded but from the 'outside' it is not.