I wasn't travelling with a baby, but here is what worked well for me. On our trip 2 weeks ago, it was just myself and my 9 year old niece. I wanted to minimize the amount of stuff to carry through the airports, minimize waiting time for an airport shuttle to the resort, and maximize how far my dollars would go.
I shipped a box of non-perishable food (also, ponchos, sunscreen, etc) down ahead of time, and planned it so that the package was scheduled to arrive 3 days before we did. That way, I was able to check the tracking number and confirm before I left home that it arrived at the resort and got the name of the person that signed for it. I figured if the package didn't get there on the guaranteed day, they would have a couple of days to locate it. It arrived on schedule, and it had slipped my mind when I was checking in to the resort, but the CM said, "we have a package for you, I'll bring it out for you or we can have bell services deliver it." Shipping the items ahead worked out great for me.
Also, I didnt need a rental car for the trip, but decided to use rental car to get to and from airport. I booked two one way rentals with Dollar (using the entertain code, I got an economy car for $30.23 (including everything) -- there is no one-way drop off fee on these rentals). I wasn't a "Fastlane" member, so I signed up for that online and then booked the reservation the next day when I was notified of my member number. At the airport, I stopped and picked up the baggage, then I walked right up to the counter (less than a 5 minute wait at the MCO counter, and there was a long line for non-Fastlane members--the counter help were calling for extra help from the other terminal), then, we walked right across to the Dollar lot (no waiting for a rental car shuttle van because Dollar is on-site) and we were on the road (in a Dodge Neon) to WDW less than 15 minutes after I picked the luggage up off the carousel. We stopped at the grocery store on the way to the resort to pick up milk, yogurt, cheese, meat, etc. Drove to out resort (Port Orleans Riverside), checked in, drove to the parking space near our room and unloaded everything. Since we had the car available for 24 hours, and didnt plan to spend a partial day at a park, we drove to the outlet malls that afternoon, then drove to Dollar's location at the Courtyard by Marriott near Downtown Disney. Dropped the paperwork and car keys in the drop box (the counter is only staffed from 8:30am - 4:30pm). Walked about 4 blocks to Downtown Disney, spent some time there and then caught the boat back to Port Orleans (could also have taken the Disney bus). Since I couldn't leave a 9 year old alone, she came along, but if you have 2 adults, you would have the option of having one of you stay back at the room to let your baby nap (or go splash in the pool) instead of everyone going along to drop off the car. On our last morning, we went to Blizzard Beach for a few hours, then caught a bus to Downtown Disney, walked the 4 blocks to the Marriott and picked up the car (again, paperwork was done in 5 minutes), drove back to POR, loaded luggage in car, spent a few more hours at the resort pool, then off to the airport. At the airport, stopped for curbside baggage check-in, then dropped the car off (took about 2 minutes).
All in all, the two car rentals saved me money, gave me some other options (grocery store, outlet mall) and I dont think they added any extra wasted time. And shipping the box of nonperishables was really helpful
..I could not have handled more luggage at the airports, and I saved quite a bit of money by shopping for bargains ahead of time and the shipping the items down. Good luck!