Unless you are one of those rare families that think there is no point in going to any park other than the MK with a stroller kid, I'm going to argue that staying in the vicinity of the BW is more convenient than staying at WL. (The monorail resorts are in another category, but they were not part of the original recommendation.)
From any of the BW resorts, including S/D, you can easily walk to/from 2 parks without disturbing a sleeping child. The Friendship launches have room for open strollers when they are not especially crowded. The launches from WL to the MK are a different type of boat, they are smaller and they require that you fold the stroller at all times. If for some reason the boats at WL are not running or are too crowded to wait for, you have to take a bus to the TTC and transfer to the monorail or ferry to get to the MK, from the BW area the MK busses are direct to the MK gate.
In several years of WDW visits w/ a stroller, I haven't had occasion to go through Epcot's front exit for any reason, certainly not to get to MK while staying at a BW resort. We take one of the busses from our resort to the MK, and it works fine. (When staying at WL we found it best to drive to Epcot/MGM; we just went ahead and paid for BW parking so that we could walk from there. It's easier to get out the back way after Illuminations.) Strollers are only a hassle on busses when you have the sort of stroller that's designed to only be hauled by car (or these days, by van/SUV, b/c so many of the new ones can't even fit in the trunk of the avg. car.) People who regularly take public transport with babies tend to buy a stroller that is well-suited for it, which is why Maclarens are so popular in New York, Chicago and San Francisco. If you have an umbrella-fold stroller you will be perfectly fine on busses, and there are umbrella-fold doubles on the market, for those who need doubles.
Another important way to make sure that your stroller does not take control of your vacation is never to use the stroller basket if you can avoid it, except perhaps to hold a lunch-box sized cooler with bottles/snacks in it. Put all of your "stuff" in a backpack or tote, and hang that on the back of the stroller. This makes it easy to take all of your valuables with you when you park the stroller outside a ride or restaurant. Just be strict w/ yourself, and avoid the temptation to drag the kitchen sink into the parks. I can fit 3 changes of kid clothes, ponchos, diapers, wipes and a changing pad, a towel, snacks, a camera, and sunscreen in an ordinary school backpack with no effort at tall, and it weighs no more than 2 lbs.
My best baby tip? Take a couple of flat cloth diapers, they have a million and one uses and take up very little space in your bag. (Among other things, in summer they are a great replacement for a baby blanket.) Ziplocs in your bag are a given.
). Tigger literally rolled around on the floor with my 3 year old once they started playing. They were great with the kids and spent quite a bit of time with us. We weren't rushed or crowded at all either. Since our seating was at 10:30, it was basically just brunch or early lunch for us. We had a heavier snack later in the day and an early dinner and did great. 
What a smile you've put on my face and quite the honor you have bestowed to me - "Queen" 
