First of all I envy you that you were able to get such a good rate. I paid $240a night and that included no breakfast. Of course this was right before Easter weekend and that suite room was the last one available (two weeks before my trip). We just ate our meals in the room, they wanted to charge $65 for breakfast for four.
The doubletree shuttle will have their first pickup about one hour prior to park opening (During our stay, MK and AK opened at 8am, so the shuttle drove by around 7:10 am, while Ep and MGM opened at 9am, so their shuttle came by at 8:10 or so; the buses were grouped, two destinations a piece, the destinations indicated in the front window). The shuttles picked up at Hilton first, then Royal Plaza, then Doubletree. By the time it got to us often 3/4 of the bus was full, but then if you are travelling at a less busy time, this may not be a problem at all). After the first pickup they would come about every30 minutes. We would catch the first shuttle to make park opening. At around 3:30 pm they switched from bringing you to the park, to pickup. This means that before 3:30 pm, if you were at the park coming home, you waited for an arriving bus to disgorge passengers to the park then hopped in, after 3:30 pm the bus would wait at the parking lot for passengers to bring home (this part we did not really try since we were at the park till closing each day), but with an infant this may be important for you.
The parking lot of the buses are furthest away from the park entrance and so their was a walk to get there, not too bad in my opinion. The buses pick up till 1.5 hours after the park closes. We always were able to catch the first bus that left the park after we got out (although we heard some people missed the first bus and had to wait another 30 min to 1 hour due to all the people- don't know if that would be an issue with you due to your time of year should not be too busy since school has started).
Overall, considering that the parking fee is $11, it ended up that we never used the car going to Disney parks. We used it only to go to the Universal Parks. So I think if you can get to the hotel some way, I may not even rent a car. In fact they even have a bus that operates in the afternoons that will bring you to Downtown Disney and pick you up also.
I hope this helps. The hotel room was nice, the beds were very comfortable. Ask me any other questions you think of.