We had anemone bulbs one year. They did ok, but didn't come back. But that was a long time ago.
Crocus... squirrel magnets! If you do put them outside, you will need to cover them with something. Especially as they aren't planted too deep in those trays. If you have any old screens, you could lay them over the top. Normally, I'd suggest the nylon net, but being so shallow, the squirrel might tear right through it. You could put the netting over them and then dump a bag or two of mulch on top. That would insulate the bulbs, but they'd still get water. And, if there's a couple of inches of mulch on top, it might be enough to discourage the squirrel.
I total get the heart-wrenching feeling that you get throwing those bulbs down the chute. I wonder, would they still get the same write-off if they were to 'donate' that kind of stuff to a farm? I'm not sure about the daffs, but other bulbs might make for good feed. (Not exactly what you want to see, but at least they wouldn't be going to the dump.)