My tech priest died on my second mission. Is he gone forever? Do I need to start over?

You must have peradeath enabled. You should get a replacement. I keep multiple saves for this issue.

I don't see any setting for permadeath. I think I have everything on default, normal difficulty.
And how do I get a replacement?

My mistake, I choose permadeath in my game. I have no idea and would assume a new priest would show up in the coharts screen.

If you don't have permadeath on (which is off by default), then the priest should still be in the same place as before, on the cohorts screen. Should have the same portrait, name, level, and augments as well.

You will have been charged a ton of blackstone for healing at the end of the mission. I assume of course that when you say "mission" you mean the entire deployment and not just a single battle - if the deployment/mission isn't done yet then you'll have to complete it before the tech priest becomes available again as they'll be gone for the rest of this deployment.

I have indeed paid for the killed priest so I was expecting to see him alive, but he is gone. He was killed in the last battle of the mission. On the cohorts screen I have now only one priest, that's why I think I am screwed.

Well that sounds like a nasty bug to me. If you have a save from before this happened, I'd reload and try again. If not, you're definitely going to have a hard time - getting hit with the cost AND having the priest dead is pretty nasty.

That said, it's probably still possible to come back from that and complete the game - it actually gets easier as it gets further along (even on max difficulty and then some), so this may actually just extend the initial challenge.

What I'd do is at least save this game for a "challenge run" of sorts, then start a new game. Also might be worth contacting the devs (not sure they check these forums), since that definitely sounds like a bug.

Note that if you're willing to be a bit adventurous and mess around with your save file, you might be able to "bring the priest back" from the dead. The save file should still have the dead priest's details saved in there somewhere - he'll just be marked as dead or something (unfortunately I don't know exactly since I've never lost a priest). It'll probably take some digging, but in theory it might be faster than restarting, depending on how far your campaign has progressed.

When I choose permadeath, I assumed it meant I'd have to replace my high level tech priests with noob priests as they died. If not, maybe I should restart before this bites me

Had that happen to me in an early mission, in a later mission I lost the entire unit - tpk style. The game kept going and I got them back, think I missed the reward for that mission rather than being given a second try at it.