Post WWII IOTL France made a huge push to be considered a major nuclear power - take a look at the number of nuclear tests they did vs. the other nuclear powers and you'll see what I mean. Presumably this was largely for national prestige since they couldn't really believe after WWI and WWII that the US would fail to defend them in a hypothetical WWIII.
Let's say instead they choose space as their prestige project. The French seem to have been pretty good on the rocket/missile technology front IOTL (Ariane, etc.) especially if they draw some support from the other Euro powers. I think the US is still likely to reach the moon first, but a three-way race might create a perceived benefit from getting there second, which could prolong the race and even put 'first to Mars' on the table.
You might see the Soviets claiming they were the best because they put the first man in orbit, the US claiming they were the best because they put the first man on the moon, and the French claiming they were the best because they put the first woman on Mars.