I wouldn't do the burpees first because they will pre-fatigue your shoulders if you drop into a pushup. A TGU needs concentration due to the complex of the movement. Also, if you have a good sweat going from burpees, the bell will be slippery, laying on the floor will make it slippery from sweat and fatigued shoulders will make it difficult. You can try it, but to me it seems dangerous.
A suggestion would be to do the getups first. Set a time, let's say 5, 10 or 15 minutes and do getups. Then, rest and try this.
Two hand swing x 10, one burpee
Swing x 10, two burpees.
Do 10 sets of swings and add a burpee to each superset so the last set will be 10 burpees. . See if you can do it without stopping. That will come out to 100 swings + 55 burpees