I have stayed at both resorts with small kids in a stroller...
Grand Floridian:
Pros: Beautiful resort. Hearing the Magic Kingdom train from our balcony was priceless.
-very peaceful
-great for kids. I don't understand the people who say it is not kid friendly. All three of mine loved this resort over AKL, which we have also stayed at.
-splash pad is wonderful and easy to keep track of my small kids
Cons:
Monorail. It has been on a modified schedule, and we had to use another mode of transportation a few times. It broke down twice while we were there. Both times we walked all the way from Conch Key to the monorail to find it was broken down, and then we had to walk all the way back to wait for a boat to get to MK. Also, to get to the monorail in GF's main building, you have to take stairs or an elevator to the second floor. The one elevator for non-main building guests fits one stroller, so there was often a line during busy times. We would sometimes have to carry the double stroller up the stairs to save time. This was a bit frustrating because we paying for the convenience of the monorail, and the inconsistency made it inconvenient multiple times. We had an impressive discount for this room, but if I had been paying more than $400, I would be pretty upset about these inconsistencies.
-you have to collapse your stroller for the GF to MK boat.
Beach Club:
Pros: one main building which was great when it rained
-walkable to Epcot and HS
-the shallow sand pool my kids LOVED. Even my 10 yr old stayed in the sand pool most of the time. It is a bigger pool area, so we just planted ourselves somewhere and stayed in that area. The 7 and 10 yr olds could roam if they wanted and the little stayed where we planted.
-the boardwalk is beautiful at night and after the kiddos were asleep, one of the us could relax at the boardwalk or walk to the IG and stroll through the countries
Cons:
-The winding halls to our room were a bit claustrophobic for me. We never had to wait for an elevator though.
- the views. We got the cheapest view room at GF and The balcony overlooked the water and the boat docks. Our standard view room at BC overlooked a roof.
-The busses. They consistently took longer than 20 minutes. We are rope drop people, and there were a few mornings when there wasn't even a CM out there to call in a bus if we waited past 30 min. Quite a few frustrated people. (This was the end of Jan/beginning of Feb). When my kids are melting, waiting for a bus is much more torturous than walking somewhere.
Here is what we did to rectify the bus situation: On Day Two, we bought the express transportation for the week. When we left a park, we would take the express transportation to Epcot and walk back to the resort. Epcot closed later than the other parks every night, so we would always hop on Test Track single rider or do a very short wait on Soarin' on our way back to the resort. Stopping for a Croissant doughnut was routine too. We did MK three days and this still worked out best for us. We would just finish up in tomorrowland, walk to the express bus stop next to Buzz a few minutes before it was scheduled to leave, hop on and be at Spaceship Earth in 15 min. Knowing the bus schedule was priceless for us. We also park hopped much more this trip because of this. The express bus was SO convenient, we easily did the four park challenge with three kids. I never would have attempted that without the express bus. We just walked back to the resort after the fourth park. I didn't miss the monorail once on this trip. The express bus was worth every penny.
All of this to say, having a toddler and stroller didn't make a big difference between the two resorts especially if the monorail is on a modified schedule and if it's still breaking down. I would choose the resort you love. My kids would tell you to pick the Beach Club because of the sand pool, but they really loved both of them.