Yes, bring a carseat... if you have one that is plane certified, use it on the plane too... We flew over 30 times before my dd was 5, and the carseat really helped her feel comfortable on the plane. Plus, there are no guarantees that any car service will have seats for your kiddo.
We usually use primetime shuttle and they are wonderful--and we get the private car so we don't have to wait on others and can go straight to our hotel. We have also just grabbed a taxi, and it was about $45 each way (LGB to a DL hotel), whereas a private car from primetime was about $100 rt.