First, leghold traps of legal size will not harm any cattle even calves, or even a pet seriously if checked daily. Very common for deer to step in yote traps and one jerk and they are gone without any harm. In 50 years NEVER held a deer, routinely release domestic pets unharmed. A 1 3/4 trap will work and rarely damages things even in the 8-10 pound range. Literally a cow wouldn't know it stepped on it. Even if stepped on perfectly it would not hold onto a hoof.
Badger are not difficult to trap because they are not trap shy and don't fear human odor, similar to catching a raccoon. A single badger will make an enormous number of holes in a small area, and the vast enormous number of them are hunting holes trying to catch their prey. If you just put a trap in the hole you might get lucky to choose a hole they revisit but you also might not and it might take a long time. Test it yourself by kicking some dirt back into as many holes as you can and see within a couple weeks how many have been re-dug. No way of knowing which hole the badger is in that particular day.
But they are easy to attract with lure or bait, especially if with recent fresh diggings you know one is actively working that area. Any kind of roadkill meat stuck well down into the hole, probably even kitchen scraps, or a few drops of commercial predator lure will bring them to the right hole. I'd cross stake (2 stakes angling different directions) or disposible anchors will hold them.
Here is a good read on trap choices.http://www.fishwildlife.org/files/Badger_BMP-final.pdf
Whatever you choose others will replace them. If you want to get rid of them, get rid of their prey.