You are stepping on a mine-field because this is one of my favorite subjects foe speculation.So, we have someone reasonably capable of administration who gets politics and is not inept in battle. But what does his rule look like versus Napoleon's?
If scenario works out, the less ambitious and less talented figure could be beneficial for everybody. Alexander hated Napoleon personally but did not have any excessively strong feelings against “republic” in general. Without the active Russian support the Hapsburgs would have a limited capacity to keep fighting against France. By character Bernadotte was much more flexible and prone to the compromises than Napoleon (and definitely not a megalomaniac so his interests would be probably limited to France) so potentially there could be a lasting peace in Europe. Domestically, he may retain some form of a republicanism or eventually drift toward a monarchy to keep situation stable.
In the area of a “personal diplomacy” he was almost definitely better than Napoleon who was using bullying as the main and almost the only method. Not only did he established friendly relations with Alexander at the moment when Alexander needed him but he was able to retain these relations during the time when he needed Alexander (and notwithstanding his activities during 1814 campaign when he went to pursue his own goals) and later establish the good relations with Nicholas I.
In Sweden he was accused in being too authoritarian and intolerant to the criticism and it seems that not all his decisions in the area of economy had been successful but he managed to survive politically (among other factors, by guaranteeing Alexander’s backing) and establish a stable and reasonably popular regime and a dynasty so it would not be a complete fantasy to assume something of the kind in France. Desiree was not a dazzling figure like Josephine but she was “from the people” (a bonus in a republican regime) and seemingly a nice person, which could add to the general popularity. Plus, the Bonaparte family was and would remain an influential political clan and their support is guaranteed.