Q39 is a nine minute drive from there and it's arguably one of the top three barbecue joints in town at the moment–this is always a hotly contested topic locally, but I'll go to the mattresses for their burnt ends and/or smoked sausage[0]. The toy and miniature museum itself is right next to UMKC campus just south of The Country Club Plaza, making the plaza a ten-minute walk. Jack Stack BBQ is on the Plaza, while not the best in town it is quite good, probably top 10. It's more about the sides than the meat if that's your thing. Their cheesy corn bake and hickory pit beans are to die for. There's also about a fifty other shops, bars, and restaurants on the plaza, too.
[0] No affiliation, just a fan.