What to say and do to encourage a depressed, chronically exhausted, chronically bitter and angry husband?

Make sure the household as as peaceful as possible, a refuge from what might be worring him. 

Don't make him do any housework.

Lose weight. Ensure he is proud of you

The part that stands out to me there is exhausted. It's possible that a return to good health might resolve some of the other problems. What is preventing more/better sleep?

Same ... exhaustion can cause all the other problems, being irritable, angry, depressed, etc.  If you're stressed out, under pressure, not getting enough sleep ... you're going to irritable, angry, and quite possibly depressed.

I think that one suggestion, if it's possible, would be to let him have a "day of rest" somewhere, like go off on Sunday after Mass or something, if he has a place to go where he can have peace and quiet, and just be alone.  You might resent that he's not around, but is it better to have him be around but grumpy, irritable, and depressed?  You need to find ways to get him some breaks to rejuvinate himself.

Also, part of being overwhelmed by stress comes from lack of routine.  I'm guessing he could use a bit more exercise also.  So perhaps he starts by getting into a routine where he gets up, and before anything else, says his prayers, and then maybe goes for a 25 minute walk to get the juices flowing.  Sedentary jobs in particlar can wear people down unless they get up and start getting a bit of exercise.