Well I see from your signature that you've stayed at Boardwalk and Swan, so I don't need to tell you how great the Epcot resort area is. I also see you've stayed at GF, so I don't need to tell you how great it is to stay on the monorail. So.....my personal opinion......we don't really care for split stays unless we're staying for a long period of time, like one week at each resort. Having said that, I can see your reasoning behind staying at Poly for the proximity to MK, and then staying at BC for the proximity to both Epcot and MGM.
If I had to choose just one resort, I would say given the ages of your kids to stay at BC because of everything in that area from the Epcot countries for meals, to the boardwalk for meals and entertainment, ESPN Club, surrey bikes, easily watch Illuminations and/or Fantasmic every night if you want. Unless of course if your kids love MK the most, then I might tend to favor Poly.
Those 2 resorts are my favorites and I can see why you'd choose both of them. We tend to rotate between the Epcot resorts and the monorail resorts. This September we are trying CR tower. It keeps things interesting to switch around.
Good luck!