Franz was no longer in Vienna, but he was still free and legitimate sovereign of Austrian Empire with full power on his army, administration and government. Occupation of the capital city did not change the personal power of sovereign, nor the allegiance of his subjects.
No Regency was possible except by personal will of the sovereign (Marie-Louise regencies of 1813 and 1814), or if he could not exerce his will (King Georges). It did apply to the whole State, not to a former province.

Would a regency have replaced the Tsar in Russia because Poland or even Moscow was invaded by France ?

More, full approval of uprising is one thing, to nominate a man to lead it is another. No Austrian official document nominally legitimated Hoffer's command. And to lead an armed force without a government commission was, by the laws of the period, subject to death penalty.

Tyrol rose up because there was a legitimate Bavarian government according to international treaty signed by Vienna. French forces were involved later.

Legitimacy, again by the international laws of the time, was a matter of recognition of territorial sovereignity between head of State, not fulfilment of people's wish.

Andreas Hoffer uprising was certainly more than justified. It was not legal.