No they did not. It would quite literally not have been on the map for them. The short simple answer is that (and I'm describing it from what would more or less be the 19th century viewpoint) some random African so called royalty meant precisely nothing in the context of picking a national and military leader who could guarantee one ended up on the wining side in a pan-European war that the French Empire seemed to be dictating the course of.

The main consideration for the influential people (and parliament not withstanding, it was generally the nobility who decided what happened) at the time, was to restore the Swedish nation to it's status before the catastrophe of the Finnish War of 1809-10. The war that broke the nation in two and lost the eastern half (i.e. Finland) to Russia. This is the key part to understand everything that happens in the selection of a new monarch and more besides. It also forms a watershed moment in Swedish history as modern Sweden now has the borders it will continue to have and little desire to change this. This national trauma influences just about everything in the consideration of who and from where to pick a new king.

-Why is a new king needed?

The previous king, Gustav IV, despite having had good relations to Russia detests the new tsar Alexander as well as revolutionary France and Napoleon in particular. He goes to war against France (losing the last German possessions) and refuses to join the French continental system. At this point Napoleon and Tsar Alexander are uneasy friends and the Russian tsar takes it upon himself to force Sweden to join the French sphere of influence. This leads to the Finnish war of 1809-10 which, spoiler, goes disastrously bad for Sweden (and later on forms a nucleus of Finnish identity). This leads to intrigue, popular unrest, a coup and the ousting of the king who wouldn't compromise one bit with the continental powers. The uncle of the king is given the crown as Karl XIII, but he has no children. So while the immediate crisis is solved there is a solution for the future needed.

-What does the new king need to do?

I'm not sure where you get your list of requirements, but am going to try and address them point by point.

The monarch must not be related to the current royal family, since they feared no matter how they chose it would start a civil war

This I have never seen suggested as a reason. All previous dynastic shifts, from Vasa to Pfalz to Holstein-Gottorp had familial connections to the crown. It is how you established legitimacy and is normally counted as a positive. The reigning king was even Karl XIII who was the uncle of the deposed king Gustav IV. Had Karl XIII had any issue there would in fact not been any succession issues at all. Relations to the royal family was not a problem per se. There existed a number of political factions pushing for various ideas of who could best fulfil the goal uniting the nation again regardless. Since the deposed king was alive and at large in Europe the threat would always exist, and in fact, be greater the further afield you would go. They had already picked a Danish prince as heir to the crown as necessary expedient in the face of imminent invasion, but he died suddenly after coming to Sweden. Suffering a stroke and falling of his horse during military manoeuvres. There were popular unrest and some feared the return of the old king or his son.

The monarch must be christian, because church and state were so intertwined back then

Legally speaking the king had to be a protestant Lutheran. As you note one can always convert. However it does matter somewhat how alien your cultural background is. This is why other Scandinavian and German relations tended to be picked. In 19th century Europe an Ethiopian king just wouldn't cut it. It would have scandalized everyone. It's not something you would consider when you just deposed your monarch because he was grossly incompetent.

The monarch must be competent, because they feared they were one or two wars away from being annexed by Russia

The monarch must not be aligned with Russia because of the previous point

It should be noted that at this time Sweden had already had a lot of involvement from Russia in it's affairs. And Russia, while for the last century had (arguably) been the main opponent of Sweden, was for once not overtly threatening. The deposed king Gustav IV actually had had fairly good relations with the Russian tsar Paul at the time. Later making war against the successor Alexander due to personal antipathy against him and especially Napoleon. At this point a smaller nation like Sweden had to consider what it's more powerful neighbours thought.

The question whether Russia or Sweden was the pre-eminent power along the Baltic had been effectively settled with the Great Northern War (1700-1721) breaking the back of the Great Power status of Sweden, despite renewed attempts in the Russian-Swedish war of 1741-43 and 1788-90 to shift the balance. One of the previous kings, Adolf Fredrik (reigned 1751-1771) was chosen due to pressure from Russia to install a monarch more to their liking after the war in 1741-43. Ironically Adolf Fredrik's son Gustav III actually started the 1788-90 war against Russia. His son Gustav IV had fairly good relations with tsar Paul (they had similar foreign policies and thoughts about the French Revolution), but not so much with the son Alexander.

I'm writing this to point out that the Russian threat at this point was not really as pertinent as it had been a century before. Russia had effectively gained all that they needed from the Great Northern War, mainly access to the Baltic sea and thus control of it's own international trade. And that Swedish kings was and had been aligned with Russia with no issues, but also gone to war despite being in a sense aligned to Russia.

This brings us to the actual needs the Swedish saw in their king that you do not bring up.

*They wanted a king who could help with regaining the eastern half of a broken nation.*

This at this time meant 2 things, being a competent military leader and having French influence/backing. Napoleon was at the height of his power and it was perceived in Sweden that the French and Russian empires were going to clash again in the near future. Betting on the French side would mean an opportunity to regain the lost lands. They were not wrong and barely a year later Napoleon was planning and executing his war of 1812 against Russia.

By taking a French Marshal as king they thought they would get French support and a proven commander.

These two qualities the Swedish would eventually (think they did) find in one Jean Baptiste Bernadotte. Who incidentally also promised to lend the Swedish state a vast amount of money (8 million francs) through his personal envoy.

The specifics of all the confused manoeuvring of how Bernadotte kinda-sorta-but-not-really accidentally became the heir to the Kingdom of Sweden is besides the point here. I would point you e.g. to Alan Palmer's "Bernadotte - Napoleon's Marshal, Sweden's King" (1990), I have a Swedish translation here, for a more in-depth look at the scheming and manoeuvres.

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but they ended up choosing a literal peasant, a son of a farmer with no prestige at all.

Bernadotte was Marshal of France, prince of Ponte Corvo, the former more important than the latter, but he was technically not a "literal peasant", he was a prince and one of Napoleon's chosen. This was about as prestigious as you could get in the Europe of 1810.

And no matter how much one might look down on “the little Corsican”, Napoleon was de facto ruler of all of Europe at the time. And as one of the chosen few Bernadotte represented a qualified candidate for king fulfilling the main concerns the Swedish state thought it had. Bernadotte also charmed most of the Swedish establishment when he eventually arrived with his cultivation and flair. His birth may have been low, but none who met him at the time would have given it much consideration.

It is somewhat ironic that Bernadotte as crown-prince did not fulfil the wishes of those who supported him with a French-backed war against Russia. Instead he allied with Russia and the enemies of France and gained the crown of Norway in personal union instead. A gain which to Bernadotte made more sense than eastern fantasies.

And yes, the whole affair would make an awesome movie. Or tv series on streaming services.