I like the idea of a Chinggisid monarch taking Constantinople...though, that would only become an option several generations down the line.
To be precise, it would be "Istanbul" even if the Russians kept calling it Constantinople for quite a while after its fall.
However, taking it would not be something realistic until well into the XVIII century: no matter who was ruling the Russian Tsardom/Empire, the state's military strength was not adequate for such a task and, again, no matter who was ruling it, there were almost inevitable engagements on the Western borders which would not allow to concentrate on this direction. Anyway, the plan would require:
(a) Complete removal of the Khanate (including Nogai Horde) as a potential danger - either annexation (as in OTL) or bottling it within the peninsula with the Nogais being either destroyed (say, by the combination of the Russian forces and the Kalmuks) or brought to the Russian side or pushed beyond the Kuban River.
(b) Securing the right flank, which means that before any strategic offensive against the Ottomans there would be a need to get control of at least the Left Bank Ukraine (which belonged to the PLC) and perhaps of some part of the Right Bank as well (or to have the PLC completely neutralized as was pretty much the case in the OTL at the time of Catherine II).
(c) Establish a reliable operational base in Moldavia/Walachia to put Istanbul within a realistic striking distance.
(d) Ideally, but not necessarily, to have the Black Sea Fleet capable of preventing the Ottoman activities along the coast.
As an alternative, to have something like the OTL Mediterranean expeditions on the steroids: more powerful Baltic squadron carrying a lot of infantry not bothering with the Greece but going directly to the Straits; for a meaningful conquest this should be going in parallel with a successful campaign of conquest along the Black Sea shore with the penetration across the Danube.
By the time any of these scenarios is possible, ancestry of the rulers of Russia would not matter.
Definitely. The Poles/Lithuanians got on Ivan IV's case just for marrying a Muslim princess. They'd milk Simeon Bekbulatovich's ancestry for all it's worth, if (when?) the need for propaganda against the Russian Tsardom arises.
Propaganda where? It would not make any difference within the Tsardom and within the PLC the Poles would be just working hard convincing themselves in what they already knew and rather meaningless because the PLC was routinely allying with the Crimea and the Ottomans.

For 2 other European powers that could matter, the Hapsburgs and Sweden, the geopolitical interests would overweight all these ancestry talk.