While not ideal, you can just play solo. Adaro as been giving me almost 100% larvling spawn rate so far, and I'm at about 15 Lichs. Though the spawn seems to be broken on a per person basis. It's not super good for a coop game, sure, but that's a nonissue imo, because that's about 5min of solo play.
People refusing to stab their Lich and making any further Lich spawn in the mission impossible is a much more pressing issue in the "cooperation" department.
The damage type is deterministically tied to the Warframe you use to kill the larvling. If you didn't already, I suggest you read https://warframe.fandom.com/wiki/Kuva_Lich#Mechanics
Nobody likes Impact, but realistically you should only aim for Toxin, and maybe Fire, if you're trying to get optimal weapon rolls, although most options aren't strictly bad.