You have two separate issues going on here:
1- working to provide yourself and you children with security
2- H not trusting you (due to no fault of your own)
1- working - In our case, my H realizes that I now need to be able to take care of myself, if I need to go back to school, or if I need to get a job, or a new job, or whatever I need to make sure that irregardless of him I and the kids will be fine. He had the chance for us to be a "team" in this area, I was a full fledged partner, but he was not, he lost that privilege of his own choice, I did not take it from him, he threw it away, as your H did. So, I see it that your H has to work on this area. All you need to do is do what you need to do, do not take on his issues, they are not yours to solve. If he acts childish, I would just ignore that, and do what I need in a polite manner.
2- trust: you have NO control over this, you not applying for a job or staying an at home mom will not do anything to help him trust you. You can do nothing to build his trust because you have always been trustworthy and he still does not trust, it is about him, not you, again, this is not your issue to deal with, it is his.
All you have to do is choose how to react/respond to his "issues" when they flow into your world, you do NOT have to take them on, you do NOT have to coddle them and so on, being respectful, but setting up boundaries so he realizes they are his issues to deal with sound good to me.
BTW I too was a SAHM and am finding it very difficult to get employment that is worth my time, even with schooling, it is very competitive out there and things have changed greatly in the last 15-20 years, if you get an opportunity, I would advise you to take it.
Grace