I wouldn't think renting outside the airport is going to be cheaper -- at the airport you'll get the greatest competition since all the companies are there. I gave up using the big name car companies years ago and nowadays use economycarrentals.com. They're a broker who sometimes find smaller local companies, although in latter years I find they're just as likely to go with a bigger company implying that the bigger company rates have become more competitive. They throw in various added extras for free (extra drivers and CDW) though I've never had occasion to test how reliable those things are. I haven't hired babyseats but I know people who have and they're expensive. Also, the perennial thing to watch out for that people forget -- even if you have CDW from elsewhere the car company will put a temporary charge against your credit card during your hire period so make sure you have the credit to cover it.
P.S. I have no connection to the company I mentioned, just have used them for a few years. And by way of disclaimer I got gouged for damages by a car company once in spite of cover, but it wasn't ecr's fault (long story).