The big technical problem in your scenario (even admitting its political plausibility) is that Jean Bart had only one main armament turret fitted in 1940 before she escaped Saint Nazaire. The four 380mm guns of B turret could not be installed on time and were left behind as the captain refused to take them on board the ship because he was very concerned (rightly so as it happened) that Jean Bart would get grounded in the Loire river estuary, as the canal that had been dredged for the ship was very tight. Jean Bart was stuck two times in the epic escape to the open sea.
I doubt that even the US war industry would have been able to manufacture 4 large calibre naval guns while in full war effort. As a matter of fact, that question was investigated in 1943 after Jean Bart and Richelieu joined back in the war and at that time it was decided that completing Jean Bart as a battleship was of marginal military value considering the huge industrial effort required. Beyond the main armament issue, even secondary 5in guns and directors were in big demand and rather scarce at the time. Alternative options were explored, such as completing the hull as an aircraft carrier, but to no avail. Instead, it was decided that Jean Bart would donate her only 4 main guns to the Richelieu so that one French battleship at least could contribute in the war.
In the very interesting uchrony
http://www.1940lafrancecontinue.org/ that is working on an alternative history based on the assumption that France did not leave the war in 1940 but battled on from North Africa, the team decided that Jean Bart was completed as an aircraft carrier.
Even in that context, the CV Jean Bart is a moderately successful ship as it can deploy about 50 aircraft, rather less than a much smaller Essex.
Even after the war, the French navy thought very hard about what to do with the ship, including scrapping her, but in the end, the heavy guns that the Germans had seized and transported to Norway to serve as coastal artillery were returned to France, which permitted the navy to complete Jean Bart to a modified design, with arguably the most advanced AA ever installed on a battleship.
This scarcity of main guns means that in 1943, there were only 10 undamaged and serviceable 380mm guns (6 on the Richelieu and 4 on the Jean Bart) available to the allies to fit to French battleships, i.e. only enough for one of them, as it happened. Manufacturing ammunition specifically for French battleships was already a big effort for the US war industry. I doubt that manufacturing heavy calibre guns as well might have appeared worthwhile or even possible at the time.
Nonetheless, in your alternative history, you are free to disregard such limitations on the main guns. If you do so, I suppose that your reborn Jean Bart would look very much like a South Dakota class US battleship, with the standard radar fit of US capital ships.
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4 Mk-37 directors with Mk-12/22 radar for the 5" battery, Mk-13(?) main battery director atop the forward superstructure. Probably an SC air search, SG surface search, and a YE Homing Beacon for aircraft. Most ships had 2 air search/surface search sets for coverage limitations and backups, especially the Carriers and Battleships.
