Unless you think there will some sort of incredibly good promotional deal when you take your trip, my advice is - go to a website that rents DVC points and get a 1BR unit at Bay Lake Tower.
OK, actually, my real advice is - go where YOU want to stay, the 1 year old will probably roll with the punches better than a kid at the age of 2.
But because of the concerns of strollers and transportation and all of that, staying at Contemporary or Bay Lake Tower, to me, makes the most sense. First of all, when you go to MK, you won't even have to wait for the monorail or a ferry; there's a 10 minute walkway. If for some reason your kid is having a really rough time or has been sick or had diarrhea and you already went through your backup outfits, you don't want to wait for transportation, and here you wouldn't. Doesn't help as much when you go to other parks, but I suspect you'll be spending most of the time in the MK anyway, correct me if I'm wrong. And if you have Park Hoppers, this makes it easy for you and your spouse to take turns walking over to MK while one stays in with the child either for naps or at nighttime.
Second of all, when you have a kid that little, I think it's good to take advantage of the interaction at character meals so you're not wasting time in the parks getting those pics of your kid and Mickey Mouse or whoever. Staying at Contemporary or BLT gets you walking distance to Chef Mickey's at the resort, plus Crystal Palace and the Royal Table in MK, and a monorail ride to Ohanas at the Poly or 1900 Park Fare at GF. And 2 monorail rides (which is an attraction in itself IMHO) to Epcot with their Princess breakfast, YMMV if you think your child will dig them over the more cartoon-y characters.
Third, if you splurge for MK views, you never to worry about missing the show at night because the wee one has to sleep. Or like I wrote above when it comes to going back to the parks, you could take turns with one watching the child while the other goes to one of the observatory areas to see the show, or get a drink at the bar at Cali Grill, etc.
FInally, if you do opt to rent points and get the 1BR unit, it gives you a little bit of space to hang out at night without having to keep everything quiet. Whether you let your child sleep in the pack-n-Play in the living room or (as we did), the bathtub area in the master bath (close to the bedroom in case of tears, but BLT 1BR units have 2 bathrooms so you're not risking waking the kid at night), you don't have to suddenly keep everything quiet until morning.
Sure, the CR and BLT don't scream Disney whimsy as much as say the Wilderness Lodge. It's more understated, if not muted, but that magic is there, IMHO.
That'd be my recommendation, but whatever you decide, have fun.