All the good advice is likely to get you fired. You're shooting spit balls at a guy with a cannon. And he sounds like an idiot with a cannon.
There are many times it feels like this. Or another way of putting it, your trying to put out a house fire by peeing on it.
UPS has done things their way for decades. Nothing new, nothing really changes, only the faces and names. While I would not classify the center manager as an idiot, he does bear watching and using careful methods when dealing with him. If the tirades keep occurring, then someone needs to file. Everyone can loose it once, but if it becomes a trend, your BA needs to step in and have a serious discussion with the DM and district labor.
Grievances don't do much to the manager. They're just a nuisance and an ego buster. Warning letters, suspensions and terminations are not an even trade in my opinion.
While I dont agree totally with his statement, there is some truth to what he is getting at. As a steward, you have a huge target painted on your back. One that you volunteered to wear. And for that, you have my respect. Many that have been stewards are not worthy of the post.
As a result, management will always keep in mind that you would be a trophy hanging from the center's conference room. Your integrity needs to be 100%. You need to give UPS 100% in all you do in your position. You need to conduct yourself with dignity and honor, despite what management does. That way, regardless of what happens, you are blameless on all counts. Not because they dont try, but because everyone knows you well enough for crap to stick. And believe me, the more successful you are, the greater the effort to find something......... Dont give them anything..
It makes as much sense as picking a fight with a cop. Good luck.