One idea for you, as we have done this with our kiddo in the past (now a seasoned international traveler at the age of 10):
Look carefully at the length of your car rental vs. what the agency charges for a car seat, it may be cheaper to actually buy a car seat than to rent! This is especially possible if public transport is available at the start of your trip.
On our last trip to Germany, I used Amazon.de to purchase an inexpensive seat and (after clearing it with the hotel manager) had it delivered to our first hotel the day before we arrived. Luckily by that point our daughter was big enough for a "booster", but given the length of our trip, even a full-backed 5-point seat would have been cheaper to purchase than the rental fee. Some big cities have charities that will accept a lightly used carseat as a donation, but for the massive cost savings vs. renting we were happy just to abandon the seat with our rental agency!
After flying with a car seat a couple of times domestically, I learned I never want to carry a seat on a plane again. It consumes a carry-on bag, since most car-seats are NOT rated for Airline use, they won't let you seat the kid in the seat on the plane! We've also seen that rental car seats are usually beat to hell, and the agency is never as prepared to deliver/install it as you would like. (Most recently on a "family" trip we waited nearly an hour for a carseat for our nephew.) Go to youtube and review carseat install videos so you are confident (and competent) to install the seat on your own, as this is not a job for the minimum-wage counter jockey.