Or, a Ottoman Sultan can convert to shi'ism.
If that happens the Sultan will get killed or get replaced.
The No Safavids is the best option since without them Iran will be easily conquered.
Guess how long Ottoman Empire would stay peaceful. If sultan tries, he will not live long or there might be civil war between Shias and Sunnis.
I'm not sure where this idea is coming from. When the Safavids came to power, most of Iran was still Sunni. Why is it any more likely that a Shi'a Ottoman would be overthrown than the Shi'a Safavids? Especially if it's prior to the Ottoman conquest of the Mamluks.
Unfortunately, Turkey as a geographical term didn't come to much until the 20th century.
This is the key issue. Turkey was not a geographic region (Anatolia was of course, but it was rather heterogenous), and Iran wasn't really either. If by Austria-Hungary style you mean what I think you mean, as in a dual monarchy where Austria and Hungary are self-governing with their own citizenship, nationality, laws etc, then it's simply not viable. a dynasty that rules both Iran and 'Turkey' isn't going to rule them as separate, distinct entities. Even in the event that it for some reason did, Iran and Turkey are not at all homogenous - you'd have to have Azerbaijan, Iraq, Balochistan, Armenia etc etc. There's just so much more to it.
On top of that, any Turkic dynasty that rules over Iran will have a population consisting largely of Persians and will more than likely focus its capital more towards Iran (I can easily see a Turkey-Iran spanning empire putting its capital in or near Baghdad), and will very quickly become culturally and linguistically Persianized, as happened to the Seljuq, Safavid, Qajar and basically all other Turkic dynasties.