Nappy can cross the Bosphorus if the RN does not prevent it. It is even ideal. Nappy caught in Anatolia with his large Army not able to cross back to Europe while Prussia, Russia and Austria take the advantage. But Nappy won't cross. That is too much of a risk and the RN will still prevent it. Besides. If the RN is near Constantinople Napoleon can never secure Constantinople
Of course, an idea of Nappy going into Anatolia is unrealistic, he was anything but an idiot, but you are seemingly forgetting that at this time Russia is his ally AND
at war with the Ottomans (occupied Wallachia and operating in Bulgaria) so, in the scenario of Nappy's joining the entertaining, it is a French-Russian race to Istanbul with the Ottomans simply not being strong enough to fight on 3 fronts (Western Balkans, Bulgaria and Caucasus).
As for the RN. The Russian fleet (2nd Mediterranean Expedition) is operating in the Aegean Sea (defeated Ottoman fleet in the Battle of Athos in 1807) and is well positioned to reach the Straits earlier than the Brits. As soon as the allies hold the European coast and Russian squadron is in the Straits (and can be easily strengthened by the Russian Black Sea fleet), it is highly unlikely that the RN is going to try to get in or to do much more than just organize blockade of the entry. Technicalities of the situation had been demonstrated at the Battle of Dardanelles (10–11 May 1807): when Russian squadron had been pursuing the retreating Turks to the Dardanelles the heavy fire of the shore batteries and darkness compelled him to give up the pursuit. However, the problem with "holding Istanbul" was not a physical possession of the city but a plain fact that it heavily depended on maritime supply. Prior to the Battle of Dardanelles Russian fleet maintained its blockade for 2 months causing food riots, deposition of Sultan
Selim III and attempt of the Ottoman fleet to break the blockade (which led to the battle). So the victorious allies, after they got where they wanted, may have a serious problem on their hands: how to fed all these Turks.
Strictly speaking "Napoleon going to war" does not necessarily imply that he is going to the Balkans
personally (see Merime's comment to his own "Napoleon and Montenegrians"

): the theater is not big enough (both geographically and geopolitically) and he can just send there an army lead by one his marshals as a show of his friendship with Alexander (IIRC, Nappy was making some noises about "sharing the world", etc.).
The most obvious and practical goal for Nappy would be to extend Illyrian Provinces further inland along the Southern Austrian border (thus creating a potential danger for them to be attacked from this direction if they don't behave). He could be sympathetic to an idea of the Serbian independence providing Serbia becomes
his (and not Russian) client. No need to send a lot of troops there (causing understandable problems with the looted "liberated" population), just a token force: the Ottomans, would have to evacuate area anyway if they don't want to be cut off by the advancing Russians.
Of course, all this brings up an obvious question: why would Nappy do anything of the kind if he considered the Ottomans as a valuable distraction for Alexander who by 1810 was visibly getting back to his traditional anti-Napoleonic stance? In OTL he was vocally unhappy with the news about the Ottoman defeat in 1812 ("These Turks have a talent to be beaten!") because it meant that Russian Army of the Danube can be deployed against him.