Hi H.! I just took a 2 hour flight + a 1.5 hour car drive following the flight with my 3 month old last week. Unfortunately, my husband was unable to join us on the trip b/c of work conflicts so I did this alone. My husband was able to drop me off and pick me up though, so that was nice. I nurse exclusively as well. Honestly, taking an infant on a flight is not that bad. There were a lot of children on the flights I was on and it seems it's the one year olds that are the most fussy. I gate checked my carseat and stroller. At the airline counter ask for a plastic bag to wrap your carseat in. That would be a cheap alternative to buying a carseat cover. That's what I did and my carseat was just fine. I didn't have a cover for my stroller. I did lose a cup holder on it, but other than that there was no damage done. On the plane I tried to nurse her on take off and when landing. She was fussy taking off and I couldn't get her to nurse, but once the plane was up in the air and really moving she went right to sleep. On the way home she had a diaper explosion - which was not fun trying to change her in the airplane bathroom - but I managed. Just be sure to bring a change of clothes for your baby and a shirt for yourself (just in case!) If you have a baby carrier such as a Baby Bjorn, I highly recommend taking that to get through security. Even though your husband is there you will need a free set of hands to take off your shoes, etc. They let me walk through security with my daughter in the baby bjorn and there was no problem. If your baby has a favorite toy - my daughter loves the soft book - Busy Bee - and I took that. On the way home she was interested in it for a while, but she mainly loved looking around at all the people and seeing the lights.
I think your car service picking you up at 7am is perfect. If you were driving yourself that would be a different story, but you will be dropped off at your terminal by the car, so 7am should be plenty of time.
Good luck - it's not that bad!!!