This is an interesting topic. I'm trying to think of the most efficient way to get up from the belly. All the images in my head aren't incredibly graceful to be honest!
Usually people arise from this position similar by starting with a seal pushup. Hindu pushups might be good to look into a la Great Game protocol. I'd also think various components of Max Shank's flow movements such as mountain climbers with a reach (world greatest stretch) and two legged mountain climbers. I'd think the burpees you practice might also transfer well, as well as pushups from a dead start. Perhaps the neglected component is the hip mobility to get the feet planted so one can stand. This is where the mountain climbers come in. Lizard Crawls, although quite tough, could be of value there as well. Basically, spend more time on the belly and get up from there. The closest thing I practiced with this goal in mind was an explosive burpee, where during the ascent of the pushup, I'd jump (like a full body plyo pushup) and swing my legs under me, landing in a semi squat and rising from there. Hope this gives you some ideas!