I have a point in my mission where an NPC contact ship and the player are ambushed by hostile groups. The enemy ships attack me when in range, but they ignore the contact ship, which in the story is the ship they've come to destroy. Is there any way to either have the NPCs attack the contact, have the contact flee to warp when the enemy ships decloak, or have the contact ship blow up when the enemy ships decloak?

With the S7 update you'll be able to hide contacts. For now, however, you could try replacing the contact with a Federation group, reskinned to be the ship you want. Then under Behavior change it to Timid Creature. Then when combat starts, the enemy ships should target the ship, which will try to run away. At least that's how it should work. I haven't actually tried it in space. I believe it works on the ground.

Or if you want the ship to fight leave it set to Legacy Combat or True Combat (I personally think the Legacy Combat/Contact is more reliable since the new behaviors seem to be kind of buggy).

Even on ground, the timid creature behaviour is iffy. Sometimes they run sometimes they just stand there

Sounds like that's the best bet for now. I suspect I'll be doing some major reworking of this mission when the Foundry is updated, since it is introducing my fleet starbase, and we don't have the foundry assets for those yet. Hopefully this update will allow us some more control over what our NPCs are doing.