Do I Kill the Monger in the Cave or Theater?

This makes "Killing the Monger in Cave" the best option which not only helps you from losing a member but it also saves you from taking hard and avoidable decisions. However, without spoiling the game's story for you to a great degree, it is advisable to kill the Monger in the Cave and not in the theatre.

What happens if you dont kill the monger in public?

Quest: Monger Down

If you publicly execute The Monger, in a quest later, To Kill or Not to Kill, you will have no choice but to kill Lagos the Archon, a cultist who is friends with Brasida.

Who should I help take down the monger?

Dodge when he glows red and use abilities to take him down. If you're lucky, a mercenary will join in the battle and help take him down. At the end of the battle, Anthousa and her hetaerae will finish him off. After the Monger has been killed, a conversation with Anthousa will begin.