Yes it sounds as if he is being managed out and he needs to keep his nose clean so whatever the minor infringement was, he needs to make sure that's the only slip up - don't give them the ammunition.
This might sound harsh, but most companies are there to make money or achieve a specific purpose and as such, they want the best people in positions. It is of no benefit for a company to go out of its way to make it harder for employees to meet their targets. It may be helpful for your husband to reframe his thinking and think about other possible reasons staffing has been reduced rather than it being a way to get at him - if the previous company was failing there is a good chance the finances or other indicators still aren't adding up.
I think it's interesting that the employees are working to rule. Sometimes this standoff gets into a war of attrition and the bosses think they have to be hard line but the more hard line they are the more resistant the staff become. Your husband sounds like he is in the middle of something like this. Is there scope of him to do some kind of mediation? What does he think the solution would be?
All this aside, if he is being managed out, he needs to decide what to do about it. Does he want to stay? If so, he needs to clear the air and find out why what he thinks is happening is happening. Or he needs to be planning his exit strategy.
But yes, keep records of everything. Document every decision and its rationale. Be punctual. Don't give them any room to sack on a technicality.
Regarding putting a grievance in...I'm not sure that there is any way back from it. If there is a genuine attempt to manage him out I can't see putting in a grievance having any effect.
Hope one way or another you and your husband find some peace - the current situation sounds intolerable 