I'm doing a Major Drills quest and my pet won't play with his toy because he already has full happiness. I took away his minion, but I don't know if he'll get sad or not before the hour's over (how long does it take pets to become sad?)
So there doesn't happen to be some way to, er, hurt my child's feelings, is there?

Tell them you never believed in them and that they are a disappointment.
No as far as I know, there is no such item. So beyond removing the minion, not sure there's much you can do.
Since I do Drills quests regularly I actually just keep a pet without a minion. It makes me feel kind of bad, but it gets the job done...
Sadly, the last time I attempted to make a pet sad it took about one point per hour. Hopefully it went faster for you though. Were you able to finish the quest? I think I read somewhere that they were speeding up the times on hunger and happiness.
sadness is about the same rate as hunger, so likely you won't be able to complete it. Drills asks for this multiple times in any set of 30 so it pays to have a pet that's desolate to start with. When I was doing Drills, I kept one of my pets out of the resort and minionless so I'd be able to feed them and play with them as he requested.