Serenitywithin,
Hi - you are certainly not being the worst husband or father. I think my dad won that award, so I can be a good judge

It sounds to me like you may have picked up a little bit of the BPD tendency for black and white thinking. Working with BPD is not an "I did it, it's done," kind of thing. It's not a "I helped my wife and neglected my kids I fail," kind of thing. It CAN be a little "I have some balls in the air I am juggling and some will be higher than others at various times, and a few might even be dropped but I can pick them up," kind of thing. It's a process, not an end.
Anf from what my friends tell me, just being a parent is like that, even without BPD.
Kids - kids need at least one stable, consistent parent. Based on their ages, you can find kind ways without bashing your wife to let them know about her disorder so they can be a little protected. Your T might be able to help with that, how to communicate appropriately based on age. Mostly, if they can know "Mommy can feel stressed at times and this can make her get mad, but that's okay because even mad she loves us and just can't show it at that time."
Its great you have a T to talk to - I bet she can be a really good resource.
Family input - I'm sorry, but your family can be the very worst to ever give advice, especially when something as tricky and nebulous as BPD is involved. Most people simply can't understand it. They hear the crazy, and say, "leave". They don't understand there's enough of a loving, kind, caring person in there you just can't u and go. I tend to filter what I tell friends or family. It does me no good to have them attacking from another front, telling me things I can't do or have no inclination to do, just because they want to feel the issue is solved. BPD is an ongoing thing, much like diabetes or other chronic, forever conditions. You don't just give someone insulin once and that's it, not do you say, you watch your sugar I'm going to be a jerk and we can all have cake. There is a middle ground, this is what we look to reach.
Stopping worrying about her feelings - this is partially decent advice, but not in the way they mean. - Her feelings are HER responsibility. Not yours. Not the kids. Hers. You can be supportive, you can be loving, but she has to manage her own emotions, and you are allowed to have your own, distinctly separate ones. I bet like many on here, you face some codependency. I do. I often feel for me to be happy everyone else has to be. If someone is sad, I'm supposed to "fix" it. I can even reset H for being upset if I feel fine because I can feel conflicted about "can I still be fine or do I need to be upset, too?" As a child of 2 BPD adults, I can say being allowed to have my own emotions is a revelation that was long coming. This is something to work on with the kids. I bet some have picked up on feeling they need to manage mommy's emotions for her.
You should not be expected to pick/choose between your W and your kids. If your W is being particularly abusive, yes, set a boundary and take the kids for a walk to get them out of the line of fire. Yes, tell her abusive action is not going to be accepted. Find ways that work for you to enforce that boundary. I'd pick one, good, easy to remember boundary to start with. Remember, it's a process. Pick a boundary, get used to enforce it. Get comfortable with it. Get the family sued to it being a new normal, a new fact. It could be no yelling. It could be no derogatory statements about herself (triggering the kids to validate her - that is something I remember, pretty unhealthy. I was mom's caretaker, confidant, emotional booster... .but she was not able to reciprocate. I was more HER mother from age 7 forward to a point where in my 20s she was like a wayward teenage daughter for me). Anyway, once you get used to enforcing the boundary, you can pick another and then work on it now that the first has been established and you are practiced in using it.
The trust comments about lying - this is tricky because she's wanting a black and white answer to a question about which you feel very grey. You DO have to trust someone to see if they can be trusted. Sadly, there is no other way. But I think the phrase is "trust but verify." Not blindly, not without doing a few checks when possible. She wanted to avoid blame (BPD symptom) by trying to shove the previous lying under a rug and ignore it. And sometimes with BPD we need to do just that, ignore some of what they say and do as irrelevant or unimportant in the big picture. There are times I know H is wrong in what he is saying, so I have to gauge "is this even important? Is this something that will affect us badly?  :)oes this need to be corrected or will it mean nothing in 10 minutes?" Knowing he will mold and shape conversations to fit his emotions of the minute, I find corecting him at tiems is useless and just causes conflict. Pick and choose the battles, as you can.
Again, you are trying to juggle a lot. It's not your job to make your family satisfied with your r/s at this time. So that's a ball you can stop worrying about - just tell them 'I am seeing a T, this is a long process, and the kids' welfare is on everyone's mind."