What we has to estimate is the influence of the French Communist Party (PCF), which plays (since 1940) a big role in the French Resistance (but unofficial because of the German-Soviet Pact). The PCF leaders are in jail since 1939-40 (they have been transfered to North Africa in FFO) and if you remember, Jacques told us in the chronology that they are released in Feb 1942 (so BEFORE the German FFO attack on Soviet Union), after negociations between the French govt and the Resistance.
Moreover, De Gaulle is politically clever, so he won't be stubborn on the LL question.