Earlier quoted context omitted.
The loophole I've successfully used in the past in Austria was to incorporate a Ltd in the UK and operate it in Austria (similarly to how, in the U.S. The free movement provision, which is one of the central pilhars of the E.U., extends to free movement of legal persons, and does not end with free movement of natural persons. The UK is no longer in the EU, but you could go with the Republic of Ireland or Malta.
Couldn't Austria come after you and say you're running an Austrian company, given you're an Austrian business with no ties to the UK? I'd recommend talking to an accountant. In Italy or Spain this wouldn't fly 100%, it would even be a criminal charge (foreign dressed companies).
They would, and you would indeed have to do accounting according to Austrian rules and pay taxes in Austria, etc. ...so you wouldn't do it for that reason.