I know this is probably going to be seen as cheating, since this is from before 1815, but it doesn't involve a state formed as a result of the Napoleonic Wars. And those that have seen my posts before can probably guess what country I'm going to mention.
That's right, it's Revolutionary Serbia, from the First Serbian Uprising. Lasting from 1804 to 1813, the Serbs had managed to establish a de facto independent state with a government and a parliament! The oft-mentioned Ičko's Peace, had it not been rejected by the Serbs in the end, resulting in them being used as cannon fodder by the Russians, would've technically made them an autonomous tributary state to the Ottomans, forced to pay a yearly tax, but would afford them in most other matters independence. The damaging invasion the lands of the revolutionary republic received courtesy of the Ottomans with the OTL end of the Uprising is avoided, and you'd have a large group of veterans who could've protected the new state, especially if Selim III/Mustafa IV/Mahmud II attempted something whilst in the midst of the Russo-Turkish War of the time.
Whether or not there is any definite potential, I'm not exactly sure. Their position would be limiting in some aspects, sure, but in some ways it could become a regional player and a beacon of hope and revolution to the rest of the Balkans, though that all depends if the subsequent Sultans had learnt anything from the Uprising. That's not even mentioning what could've developed within the republic itself.