That's sad that you have to take a course to get a permit just to use a trap. Well that's a hassle but at least you get to use one, here in Colorado they outlawed all legholds, snares and conibears.
When I use to trap beaver. A couple of my favorite sets was setting a 330 in a bank den or setting a 330 in a good run with a dive stick or log, just make sure you stake them with dead sticks or logs or old beaver cuttings.
Sorry I haven't found time to reply... As the largest available, the 330 is used for big beavers. 280s and 220s may work just as well, but the 330 is absolutely nice when dealing with large beavers. I seen too many supers to count caught in 220's. The traps are past down through the family, so you work with what you have... Are you guys setting in lakes or rivers? Both? Are you setting in any banks? Hope to here some different technics you guys use... I have my fair share of tricks to share...Don't get me wrong there Night rider I didn't say you couldn't pull off some tricks. Just trying to help you so you didnt have to wait around for them... If I shot beavers with a 270 I wouldn't have any nuisance Beavers..I don't miss...
I can give you the recommended traps for Ontario. The coyotes we have and the few bobcats we have are similar in size to what you folks have. I think the traps you used are just a bit on the small size for coyote. Here its recommended at the very least a No 2 coil spring for coyote and you can go with a No 3 double spring a well. Anything larger will catch but isn't needed.
As for bobcat, a No 3 or No 4 double spring because of the size of their feet. Its important that you use a soft gripping trap (one with weak springs) so the chances of wring off are lessened. Usually you catch an animal by their toes because of the set placement or too small a trap.
As for snaring bears, build a cubby around a solid tree in an area where bear sign is common. The cubby forces the bear to approach at a certain angle but you still have to place the snare off to one side so that he steps in it. Wire your bait securely inside but near the back of the cubby so that it can't be carried off by other critters. Attach your snare to the base of the tree so that he can't carry your snare away. You'll find out bears are much stronger than you could ever imagine.
I have always looked for ways to save money and gain skills. I have the firearms basics covered so I rarely buy firearms these days. But I do watch for bargains. Sometimes a little TLC with simple tools will let you bring an old gun back to life without investing a lot of money. One such find was a side by side 12 gauge marked “W. Richards” It is not from the fine English gun maker Wesley Richards, it is a cheap knock off copy probably made in Belgium about a hundred years ago...