Your 3/64" is light ,but It will hold Most beaver.Most beaver dont fight a snare. Then others will just tear up the whole area.You will have a problem making a large loop.
The 1/8" will hold just about anything in North America. But beaver that have been messed with may avoid it.
I think you should make mink/muskrat, rabbit and squirrel snares out of the 3/64" cable. And build drowner cables out of the 1/8" cable.Trade of sell them off. Then get some 3/32" cable for your beavers.