I just had a player on my server lose a baby Ice and Fire dragon because they can't get it out of the pet carrier, and while that's another issue I'll bring up when I have replicated it with logs, it'd be nice to have the option to flat-out prevent select mobs from being picked up, just to be safe.

There aren't any needs in it. Pet carrier already should be available only for special list of creatures: wolves, sophisticated wolves, cats, chickens, rabbits & parrots. Any modded mobs should not be catched without mod support. So you "catch a bug".