You are fine taking pram (2 pieces) and car seat. They are allowed 2 items.. just be sure to get both pieces tagged for buy a bag to put pram/ car seat in as still very new! The bassinet is your best bet for sleeping too! Just put clothes and nappies and everything else in your luggage… time to forgo a few outfits now 😂 could those driving take your car seat or anything you might need?
Can you hire a cot where you are staying? That’s one less thing to carry. We always take a car seat when we go away as always hire a car, we put it in the hold but it does get a bit battered (ours is covered in scratches). You can hire them from the car hire company but you obviously don’t know what you are going to get. You can take your full bassinet pram in the hold (but you will have to pay extra if you take all 3 (big pram counts as 2 pieces and car seat 1 so that would be 3 ‘bulky goods’. You can get pram bags (I’ve not used them) to protect it. Have a look online. I’d take baby through airport in a carrier, as I’ve always found that way easier than trying to get pram etc through security. I’m sure your hotel/ villa / place your staying has a laundry service / washing machine - I’ve always just used that rather than taking a ton of cloths. You are in France so you won’t be far from a big supermarket where you can pick up nappies, wipes, bags, formula if you need it etc.