I've cut a lot of a lot of burl caps standing on their natural edges - small to some that wieghed over 50 pounds, here's some advice: (sorry that I can't give more specific information)
Find the flatest or most stable edge of the bark, find a VERY good place on the burl that will let you get a powerful grip, go slow, make sure your bandsaw blade is very sharp, you don't want to be using one with broken teeth.
I don't know the size of your burl caps, but here is something to consider:
The fence of my bansaw has a sturdy, straight slab of wood on it, with the width of the slab about 4" to 5" inches. This means there is more surface to press the burl cap face against.
Be careful! The bandsaw has to pass through the wood so fast and clean, that the pressure of the burl cap against fence has to overcome the drag from the cut, if the drag becomes stronger than your grip, it will yank the burl down and possibly out of your hand.
The hardest part of cutting the burl will be at the point that the bandsaw first meets the burl, and just before you finish cutting through the cap, the mid point of the burl cap is fairly easy.
Strong unwavering pressure on the fence, sharp blade, let the burl sit on its flatest edge.
Cutting burl caps reminds me of lifting 500 pound logs into the truck with my brothers and Dad: you can put all of your weight on the log to stop it from falling, or you can jump back to a safe distance, when you spend enough time lifting them, you'll first lose the fear of them falling on you, and next you'll get good enough to lift them the first time without a problem. If you think you're losing control of your burl cap, you have to forget about jumping back and instead put all your weight on it, or your not going to get your work done.