Wouldn't be too hard. I believe the Archduke Ernest stood for election and in 1587 the Archduke Maximilian was elected alongside Sigismund Vasa, and attempted to win the throne with military force but was defeated in 1588 and forced to relinquish his rights to Poland in 1589. So it's certainly possible.
Perhaps in 1573 the Archduke Ernest is elected instead of Henry of Valois, backed by Spanish money who see him more agreeable than the French candidate. Henry was mainly elected because he was a more agreeable king to the Turks, and would've maintained the present Polish alliance with Turkey; however, I don't see it to be a major stumbling block. If Ernest also agrees to marry Anna Jagiellon, he may have enough support to mount the throne.
Ernest would probably put Poland close to Austria, not unlike the Vasas, and would probably support the Counter-Reformation. He may have to agree to something like the Henrican articles, but perhaps with Habsburg aid he would be able to compromise with the Sejm. Spain was very keen to see a Habsburg elected to the Polish throne in this period, as it would create an axis of Habsburg monarchs in Madrid, Vienna, and Krakow. He may not have any heirs with Anna Jagiellon (she was 50 in 1573, so is probably past childbearing: they might be able to have at least one child, but it'd be a miracle, certainly), but if he lives long enough, Anna Jagiellon will probably pre-decease him and allow him to remarry and have children: if Ernest is King of Poland and has any sons, one of them would be a natural contender to succeed him.