There was actually two French "Navies": the Vichy French navy and the Free French navy.The Vichy French Navy was probably "neutral" vis-a-vis the US Navy until the landings in North Africa but would have had a lot of resentment against the Royal Navy especially after Mers-el-Kebir and Dakar (I would not even think of Trafalgar
). The Vichy French Navy fought against the US Navy in November 42 but AFAIK it did not fire any shots against the Germans in Toulon.
However, the Free French Navy was definitely and unequivocally on the Allied side and even the British side at Dakar so the other Allies could have trusted it.
There was only one official French Navy [Marine Nationale] and a very small bunch of "dissidents" which was the FNFL (Force Navales Françaises Libres). It was the same for the Army and Air Forces. As expected, the military acted as they were ordered to do so by the French Government, which was in Vichy, except of course the dissidents which were considered like British mercenaries, Communists and adventurers. In one case (North Africa), they were ordered to shoot at anybody trying to seize any French territory, and this order was fully maintained until November 1942. On the other hand (Toulon), they were left without order beside Darlan's secret instruction to scuttle the Fleet if someone attempted to take it. As the French admirals in charge were stupid, they did not even tried to escape Toulon before it was too late. At this time, the state of confusion reached its peak and the political situation was anything but clear. Who are the official French authorities? Which one is a genuine order? What really want the Maréchal us to do? etc. Some ordered to oppose the North African landings, some did not, and the Navy scuttled most of its ships.
Now, if one consider that the FNFL (and the Free French) were more trusted by the US authorities than the officials of Vichy, it is just the opposite which happened. De Gaulle's movement was at its lowest, both in Washington and London. The Allies entrusted the Vichy officials with administrative powers (not De Gaulle) and the (ex Vichy) Army, Air Force and Navy was rearmed in priority before any 'Free French' forces. Most of the French ships were sent to the USA for refitting and modernisation with high priority and the 'Free French' units were asked to end the separation. After November 1942, the Free French movement virtually ceased to exist and was asked by Washington and London to end the 'dissidence'. Of course, De Gaulle and its followers, with very good reasons and against Washington will, were not disposed to end here the political struggle and will act in order to eliminate, one by one, every compromised Vichy official inside the reunited 'French National Liberation Commitee'.
Now, concerning the Béarn, she was disarmed as an aircraft carrier at the hostilities outbreak, and she was not worth the rearmament as an aircraft carrier; there was already plenty of new CVEs in the production pipeline and the French Navy lacked any trained air unit to complement a CVE. As Mescal posted above, her usefulness in the form she was used was certainly much better fitted with her capacity and those of the French naval aviation.