Correcting children’s behavior when husband is parenting?

Hi,
My husband is “onboard” with TSG but in reality is not doing it at all. The kids and I have a good thing going when he’s not around, but when he is, he’s impatient and inconsiderate and tends to rub some of their personalities the extreme wrong way with the unkind ways he corrects them. The children are very respectful to me and pretty much all other grownups, that has become their normal, but my husbands abrupt disrespect towards them at times brings out the strong willed rebellion in the younger ones who haven’t yet learned how to master themselves in that type of situation. An example might be getting ready for church and telling a child to tuck in their shirt and when they are reluctant, using an angry voice, yelling, and finally grabbing them strongly and forcing it tucked for them and saying how messy they look with it untucked and continuing this anytime it naturally becomes untucked prior to leaving for church. This bothers me in many ways and we‘ve had many talks about how ineffective that style of parenting is long term. As a result I’ve ended up taking on a moderator role – often telling him to get himself calm rather than blaming the kids, and sometimes verbatim giving him the words for corrections. If I even begin to give him the words to say, my kids snap into gear BUT my husband often argues in the moment that he’s done nothing wrong – so it’s impossible to help him if he’s not even willing to self govern.

I don’t want to be the moderator for my husband, especially if he’s not teachable, and so I’m planning to have a discussion with him in our next couple meeting on Sunday about roles and that I’ll be there as a resource but that I don’t want to be put in a position of taking sides.

My question is, do you recommend that I correct the children’s behavior when they react negatively to my husband’s parenting methods ? For instance stepping in and starting the rule of 3 if I see them out of instructional control with him or giving extra chores when they don’t accept his no answers etc? I know HE’d love it if I did that, but I feel like it’s his role to correct in those situations. I want to encourage them to obey him but I don’t like correcting them when he’s clearly out of line, just as much if not more since he should be the leader.

I’m sure I’m not alone in this dynamic, so I’d love any advice you have specifically for what would be a helpful way to respond when he’s escalating things with the children. TIA