I would not suggest that you spend the money on a new camera, just to help with this issue. I have done fight scenes, using a webcam, and had them turn out well. It just takes a lot of patience and practice. Even if the figure falls over, just pick him back up, and figure out how to perfectly balance him. Another tip that is key to fist fights, is to ease in and out of movements. It can also help to follow the movent with the camera. Both of these tips are better explained in tutorials by Fancypants. I'll add links a little later.
Not sure if this is the best answer, but hopefully, it can help.

2nd place Darkness and Light entry: The Tree
Behind the Scenes

You don't need to get ultra close to have a good fight scene, it might be hard, but using wider angles is an option and if done right can actually be more impressive, I actually suggest watching kung-foo movies and stuff that features hand to hand combat, watch the video essay I posted and think about timing and reactions from the minifigs. if you take a look at Grace often the most impressive bits of fighting are done in wide shots.