Sounds like you may want to use a bigger clamp
to stay away from the saw blade.
I always put sand paper on the inside of the camp and then sit it flat on the bench and hold it down and then screw it down on the stock.
Once I have it locked and it's flat I move over to the band saw and make the cut.
The band saw as no mind of it's own it will just cut anything that you push into the blade.
If the part is to big to clamp I ALWAYS use two push sticks , but I never push stock with my hands on a band/meat saw, I can replace a push stick quick and easy.
I saw the guy on American Woodshop making wooden boxes/cabinets from a blank block and it made me turn my head, he put his hand on the side of the block and push the block by the blade that was 1/2" from his hand,it just takes a sec. for the blade to snap or go off track,,,,,,I can't think of ANY wood project worth cutting off body parts for.
Wood Band saws are the only tool that don't come with any safety guards other than the blade guide so to speak and the blade wheel cover(s)
I do have a light on my band saw so I can see every thing.