As a general rule, if someone isn't handling a problem very well, it means the problem is exceeding their capacity to cope with it. You can deal with this in either or both of two ways: By reducing the demands the problem is imposing on him, or increasing his capacity to deal with it.
You say your manager is generally a good person, just not under stress, so some of my suggestions are going to assume he's willing to cooperate with you in addressing this problem. If he won't cooperate, you might be out of luck, but given your description I think cooperation is likely if you ask nicely.
Variations on your current approach
If you aren't doing so already, providing the strengths and weaknesses of each of your solutions, as well as pointing out what seems the best and why, would reduce his stress a little. The less he has to think about it, the better. Try to focus on external details as well, rather than internal ones. He likely doesn't care whether the transistor tester needs a 5v power supply or a 7v one. He just wants to know how accurate they are, what the throughput is, and how much they cost. Cut out any details he doesn't need to worry about so he'll have more energy to focus on the ones he does.
Be prepared with something else of importance you can do for a little while. This way, once you've told him about the problem and your suggested solutions, you can offer to go work on the alternative thing for a little while and give him some time to get the other disasters under control long enough to think about yours.
It may also help to print out a summary of your analysis so he can read it. Even if he decides on the spot, it can still be helpful to have it all there in front of him instead of having to juggle everything in his head.
Most importantly, talk to him like you're in control of the situation. If he sees you're stressed, he'll take that cue and get stressed himself. In contrast, if he sees you're confident and in control despite the crisis, he'll start to relax a little. Everything else might be falling apart, but at least your section of the problem is under control.
Finally, try to report on some progress every time you bring him a problem. If possible, tell him about another crisis you've addressed and how things are going well for that part of the problem now. It's easy to see all the new problems coming in, overlook all the old ones going out, and get overwhelmed. If you take the time to draw his attention to the old ones going out though, it will help reassure him that progress is being made and the crisis will end.
It might take a while for these strategies to start having significant effects, but they should all help reduce the demand on your manager. If he doesn't start encouraging your coworkers to use them too, you can share them yourself. It sounds like they'd be interested in better ways to manage their manager.
Bringing up his mood
It's hard to go wrong with an "I" statement, delivered without blame, in private, when he's in a peaceful mood.
Boss, I get pretty stressed when the manufacturing people mess up. I get even more stressed when I see you lashing out too. Even if I remind myself it's the manufacturing people you're upset with, it's hard not to feel like you're upset with me too, or like I'm making things even worse when I have to tell you about another important problem.
Keep focused on your feelings and how you're affected. No anger, no blame, no judgment, just statements about what happened and how it made you feel.
Suggesting constructive outlets
If he seems open to suggestions after the above, you might try offering some. I'm personally fond of taking a short break periodically. It might seem counterintuitive to take a break in the middle of a disaster, but stress is just as much a crisis as everything else and needs to be managed too. So here's a break suggestion:
During crisis time, take a ten minute break once per hour. Interrupt something right in the middle if necessary. Go in your office, put a "Do not disturb" sign up, close and lock the door, and just sit in silence staring at the wall and taking deep breaths until the time is up. Or lay on the floor staring at the ceiling. Maybe play some relaxing music while you're there. Try to relax all of your muscles. After ten minutes, go back out to manage the disaster again.
You might also check out relaxation and grounding techniques for more suggestions.
If your boss will agree to it, you could plan on employees reminding him to take some time and relax if he forgets and they see him getting too stressed. Be very careful to make it a friendly reminder. It'll be coming when he's in a bad mood, so it's important to be especially careful.
You seem stressed again. Are you all right? Do you need to take a minute and go recoup?
Make sure to accept his taking a short break wholeheartedly, without any negative feelings toward him. He needs to know it's acceptable to take a short break and recover, even in the middle of a crisis. Keep ill feelings down by reminding yourself that his taking a break makes him a more pleasant person to work with.
In fact, if things are this stressful, it may be worth proposing the whole department take a ten minute break every hour to keep themselves going.
Improving crisis management
I admittedly have little experience with manufacturing, but having everyone go into full-on panic mode every month seems unusual to me. If things like this happen frequently and everyone is still panicking anyway, it suggests to me that the systems in place to handle them aren't very effective and they aren't getting better. At the very least, you should have action plans in place to guide your response when things like this happen. They won't eliminate the stress, but the more organized you can make the chaos, the less stressed out everyone will be.
It may be worth taking some time during the calm to investigate what happened and why, and see if there's anything you can do to make future disasters less likely or less disastrous. Write up reports so you can examine a bunch of them all at once later and look for patterns. See if you can predict them and do some of the work early, or at least prepare yourselves emotionally.
In line with reporting progress above, you can also try to analyze the problem as it happens. Get a basic idea of what needs to be done and make a list. Add new things as they come in and cross off (but don't erase) ones that are finished. Then everyone can look at the list and see that while new problems are coming in, old ones are going out and the list is getting steadily shorter. In time, everyone will start getting used to the disaster rhythm. They'll understand that things get progressively worse at first, but then eventually peak and start settling down. The details vary, but the overall rhythm is predictable. Predictability is good for stress.
Another thing to consider is whether team members could be given more autonomy to make decisions themselves during a crisis. If your manager is getting stressed, he's probably feeling overwhelmed, so being able to distribute some of that stress to others who aren't so overloaded would be a big help.
Celebrating
People tend to forget that once the crisis is over, it turns into an accomplishment. You did it. You deserve recognition. Congratulations.
Bring in a box of doughnuts (or something similar your department all likes) the day after and have a celebration with tasty treats. Point out some notable accomplishments if you can, even if they seem a bit mundane.
I'd like to recognize Alice for discovering the broken plate on the roller, Bob for rushing to the hardware store for replacement screws, Cindy for courageously holding the plate on in the meantime, Dave for distracting the inspector to buy us time...
It might seem a bit silly and weird at first, but it's important to recognize that you succeeded where most people would have been overwhelmed and panicked. Anyone can succeed when things are going well, but it takes a skilled team to pull it off when the world seems about to end.
