Simple solution (if it doesn't work like this already): Apply the standard damage resist to these attacks. At tier five, I think that's an 80% reduction.
At the very least, THIS. I had an Explosive Esper get killed by a Rock Throw that bounced off of a robot. It's explosion hit about 5 native monsters and insta-gibbed my character. Not fun.
However...as a means of showing a character that they shouldn't attack a "Native" monster, you should have the attacks bounce off of them (Like piercing attacks do on Skelebot Grunts) in addition to the damage.
Also, some sort of "Monster Encyclopedia" where you could look at the different monsters and see what time period they're from, resistances, some background on them, ect...would be really nice and add to the game's lore.
Heck, an encyclopedia about the different time periods that you fill in as you get further into the game would be awesome. And no, I'm not talking about the "Clues" system which, IMO, works terrible.
In fact, this could be the answer to your "Bonuses for killing monstes" thing: Give the player lore entries about the monster and the monster's time period after killing so many of them. It'd be giving a reward without making it required grinding.