French buttercream contains egg yolks that have been cooked via hot liquid sugar water. How long can I safely store this in an airtight container at 0ºC?
According to a recipe from Serious Eats, French buttercream can be refrigerated for up to 2 weeks and frozen for up to several months. For the best texture, warm to 72°F and re-whip in a stand mixer before use.
This is slightly at odds with the FDA recommendations suggesting that cooked egg dishes be refrigerated and used within 3-4 days. It is possible that something else in the buttercream inhibits microbial growth, allowing longer storage, but I don't want to speculate. To be conservative, you should follow the FDA guideline or freeze immediately.