For WL and strollers - I heard it depends on the boat size and how crowded it is (and maybe the captains, not sure) - the larger one will let you roll on, but the smaller ones you'll have to fold. Regardless, you'll be folding down a fair bit. However, it is an awesome resort!
For sheer convenience of travel with little kids in strollers who need naps - I have to say BLT/Contemporary wins hands down. This is based on our stay this past Oct with our just-turned-1-year old and our almost-4-yr-old. (The usual negative is the non-kid-friendly theme, but I must say - we did a split stay and started our trip at Kidani, and my almost-4-yr-old who usually loves animals couldn't have cared less about giraffes/zebras/feeding time/birds/antelope outside her window no matter how excited mom and dad were about it. She freaked out in excitement that there was a Mickey-shaped splash pad at BLT she could see from the window and "loved the new hotel best". You just never know with kids that little).
Since we did a split stay and did the bus-stroller wrestle for a few days, by the time we got to BLT and were just walking strollers - loaded with kids and stuff and extra waters and whatever - to either MK or Epcot via monorail, it was great! It was even better on the way home when the stroller nooks and crannies were full of half-drunk sippies and random snacks and toys and wipes and etc. We stayed up for parade/fireworks one night, and to take the sleepy kiddo and put them in stroller and walk that all the way back to our room past all the crowds with no interruption was awesome. Naptime could start in the stroller and finish in the room and it was no problem. We had plenty of touring time and we took 3 hour breaks in the afternoon. (We'd get there early and tour through lunch, then break in our room and return to MK around 4 and leave again around 7:30 for our usual 8pm bedtime). We got plenty of touring time in and we had that much-needed afternoon downtime.
Also we thought we'd end up going back to Epcot, DHS or AK towards the end of the week but the kids enjoyed everything at MK so much more we just kept going back.
Kidani would be very car-friendly - you can park under the rooms, so you're never that far. If we had to stay there again, I probably would rent a car. But you're right, parking at MK - ugh. While the pool area is fantastic, it was sort of wasted on our 4-yr old - she was just as happy at the BLT pool. When she's in that 6-7-8 range, Kidani pool will be a bigger draw I'm sure. However, we adults definitely appreciated the resort and animals and theme more than either kid by a long shot.
Good luck, hope this helped somehow!