Why would you walk all the way to the front of Epcot to catch a monorail to Magic Kingdom, when you could simply take a bus straight from the Beach Club to the Magic Kingdom?
IMO, Beach Club all the way. 5 minute walk to Epcot, about a 15 minute walk to Hollywood Studios, an easy bus ride to Magic Kingdom and Animal Kingdom, and the best pool on property. Not to mention a ton of phenomenal food options throughout the Epcot resorts.
You're right, you don't need to go from Beach Club to Magic Kingdom by monorail. However, if someone's park hopping between MK & Epcot, it's feasible to use the monorail one way or the other.
Contemporary or Beach Club both make wonderful choices. You're right in that Beach Club has many dining options within walking distance between its own restaurants, Yacht Club, Boardwalk and Swolphin. But being at Contemporary also affords you a lot of dining options. Not only are you talking about 3 very-different-but-all-great restaurants in your own resort (and slightly off topic, my kids love "monorail-watching" at the Contempo Cafe as well as Chef Mickey's), but you can ride the monorail to any of the restaurants at Polynesian or Grand Floridian (and the Magic Kingdom, of course), plus you have boat access to Wilderness Lodge and Fort Wilderness, making it easier to get to WCC or Hoop De Doo.
What would make me lean more towards recommending the Contemporary to the OP would be:
1: the plans are for this trip to be in January. It's extremely possible, perhaps even probable, that the weather will be accommodating for swimming in January, but the chances of a cold snap are there. Plus, while everyone's mileage may vary, I know that if I only had 5 days at WDW, I'd be as much of a park commando as my and my family's stamina would allow, which might afford very little time to swim.
2: the OP's kids are young enough that a lot of time would probably be spent at MK. Or if they have park hoppers, it'd make it easy to go to one park and then decide to hit MK for one or two favorite attractions before going back to the hotel. I suppose you could do that with Beach Club as well, go to MK or AK in the morning then bus over to Epcot or Studios for a brief visit before going back to the room, but I'd hazard a guess more "favorites" might be at MK, and the trip back would be faster and easier. But YMMV.
3: The OP already stated the restaurants at the Contemporary are amongst their favorites. With only a 5 night trip, make it as easy as possible to go to your favorites unless you really want to branch out and make yourself try new places.
EDIT: Not that big a deal, and the OP's kids might be different, but I have kids who are currently 8 and 4, and for them, one of the things they're most excited to do is ride the monorail. Last time we went, we stayed at BLT and had a car. We drove to Epcot, but I deliberately kept my oldest daughter there for "just one more ride" when it was time to bring the youngest home for a nap, because then she could have that long ride back on the monorail which was way more important to her than the Malestrom (she did not care much that the ride was about to close and this would be her only chance to ride it). Anyway, my point is, we rode the monorail as much as possible. Even being within walking distance of MK, we still sometimes rode the monorail to MK and ALWAYS rode it back because the kids wanted to so much. I also tried to book as many meals at Poly & GF as I could to make it necessary to ride there. Don't sell the allure of the monorail short with younger kids.
