I was shocked when I saw the title because this was something I've been thinking about for some time. In OTL, in 1866 after the palace coup against Cuza, the throne was offered to Phillip of Flanders, a relative of the Bourbons. Fearing that he may upset Napoleon III, Phillip refused through the General Consul of Belgium in the Romanian Principality, Jacques Poumay. Then, at Napoleon III suggestion, the throne was offered to Karl of Hohenzollern-Sigmaringen, who accepted and became known as Carol I, Prince of Romania and then the first King of Romania, after the Romanian War of Independence (Russo-Turkish War of 1877–1878).
So to the OP, yeah a Bourbon came pretty close to becoming Prince of Romania (and later even King maybe if all goes well for him). And to The Ubbergeek, there's no need for Romanians to turn to Catholicism ...
Except the Belgian was a Saxe-Coburg-Gotha. Yes, the Belgian Royal House had Bourbon blood (that of the Orléans branch, not the direct Bourbons) through Leopold I's Queen, by dynastically they were of the house of Saxe-Coburg-Gotha, who supplied many Balkan nations with their kings. We need a French prince to be offered the throne, probably one of the sons of Louis-Philippe. Or alternatively, the 1815 Restoration endures or the Duke of Berry gets a more lucky, so there are direct Bourbon princes floating around.
I think I speciffied he was a "relative of the Bourbons". And a French Prince being offered the throne is not so hard because Romania was very Francophile at the time, especally the nobility. French was like their second language ... maybe even their first.
I think a Spanish Bourbon would work for Romania, I think Transylvania given to the Bourbons would be a good start.
Aren't the sons of Louis Philippe also Orleans ? And the question is even if they are offered the throne, would they accept ? Even Carol I was reluctant at first (and he was only an officer, not the son of a king), as Bismarck advised him at first to refuse because "Romania was more of an Asian Kingdom than an European one" (quoting from memory), but then he advised Carol to accept telling him "it would be a great adventure of your youth" (also quoting from memory).
The House of Orléans is a cadet branch of the House of Bourbon. The hard part is getting them to accept, but it wouldn't be too difficult. There are also the Bourbons of Parma, who in the late 1860s would've been recently deposed from their Italian duchy. So there is another source for a King.
The main question is how acceptable are they for the European Great Powers especially Russia and Austria (since they both a special interest in the region).
I think Romania,Naples and Sicily under Bourbon-Two Siciles can make a good kingdom and can claim as the Byzantine successor.
The Ortodoxy for the heirs is a must, it can't be done without it. The Romania-Two Sicilies stuff ... that's just not going to happen. I would be more interested in a Romania-Bulgaria union under a Bourbon ... it could be done, and even in OTL Carol I of Romania was proposed as Prince of Bulgaria at some point, but I don't know any other details or why it didn't happened, maybe someone with more knowlenge of Bulgarian history can enlighten me on this one.
Edit : Found some canditates for the job : Robert I the Duke of Parma, or his brother Henry, Count of Bardi (Bourbon-Parma); Louis, Count of Aquila or his brother Francis, Count of Trapani (Bourbon-Two Sicilies); and my favourite candidate Henri, Duke of Aumale. Some of them need different PODs to become available. What do you think ?
Are you planning on writting a TL, Kasumigenx, or were you just curious about this ?