I didn't want to take my more expensive one and have it bumped around + I worried if I took the one out of the car they'd lose it on the way back so preferred to take a seperate one in a travel bag from Amazon. I "padded" it with nappies (very expensive vs UK!) and extra blankets. Drop it in the oversized luggage belt. Never had a problem . Even Ryanair will take them free so it's worth it vs dodgy & expensive hire car ones.

Another option is to hire one along with anything else you need from a local company who will meet you at the airport