We were able to get our luggage and 4 adults into a cab. You can look for one of the minivan type cabs and you should be able to try to get you luggage in before you go. If you can't get all your luggage in, two cabs are cheaper than the 70 euros from the car service you contacted.
Has there been a change from this PDF, dated October 2006, saying €30 from Ciampino and €40 from Fiumicino? That's what shows when you search the Comune di Roma site and on the airport site in Italian.
I am repeating myself a little here but wanted to give another thumbs-up for Rome Shuttle Limosine. We just used them two weeks ago to get to and from FCO. 35E for 3 adults, 3 26 inch bags and 3 maximum size carryons. The car was just for us, spotless, smoke-free, and had plenty of room. They were waiting when we arrived and were a little early picking us up at our hotel for the return. I would definitely use them again.