Nicole: we recently bought a chair with us rather than pay the rental fee as well. Some of the rental agencies cap the max charge though so may not be $150 through all of them even if the daily fees up to more than that.
I would advise if you choose to bring you own you do what we did and discard the chair when we dropped off the car. Traveling back with the car seat was not worth the $50 in my opinion and I viewed it as a rental cost paid.
The other important thing to know is your car seat will be ILLEGAL and the seats may not have the back Latch if you child is still rear facing it may not work. In Europe they use a different system and different licensing body for the seats, not that police would ever check but using an illegal seat has some risks even if small you should know of them.
As for closing a couple of hours for lunch, you will find that true at even the larger cities as well, maybe the airport places are open but probably nothing else so yes you do have to factor that in as well as it being tougher to get gas as an American during this lunch time. Have to find an automatic pump and pay cash since your credit card won't work and gas stations will be closed as well. You will end up overpaying since change is not dispensed and the car needs to be full.