That really is weird that she's pushing for this, as I'd feel just like you! ! I suppose I am outnumbered in my opinion that they don't have to meet right this minute. And I certainly don't advocate lying to a child about why Daddy has to spend so many hours away, and I suppose that even waiting a few more years would involve having to give some sort of reason. It's definitely better to tell the truth.
So if you decided to let your dh take your children along the next time he takes his other child for an outing, do you think you could trust him to honor your feelings if you didn't want your children going into the other woman's house and spending any time with her? Just taking them all out somewhere and then dropping the other child off and coming back home? Or is Ow also pushing for your dh to take his child for overnights? And is your dh financially in a position where he can do all the overnights in a hotel, or maybe at his parents' home if they live nearby and would like to have a relationship with her?
I could see why the kids might need to at least meet each other and periodically get together. And while I don't think you should give your children up for the entire visitation, maybe you could drop them off with him at the hotel or inlaws' home for a few hours and go meet a friend for lunch or do something else just for you, and then pick your kids back up and go on home. Assuming that he'd be supportive enough to help them through the transition process, and not the sort who would capitalize on them possibly trying to push to stay longer and longer if they were having fun.
If this happened in our economic situation, which is extremely tight, I don't know what I'd do because we'd be overwhelmed by the expense of the child support in the first place, and dh certainly wouldn't be in a position to take his other child out for meals and other costly outings for every visit or spring for hotel rooms a couple of times a month.
As to the poster who likened the situation to someone's dh finding out that he'd previously fathered a child in the past, I just wouldn't have all those loaded emotions to be dealing with, in that case.
OP, here's to hoping and believing you'll find that stream of light that can guide you and your kids through this dark time and into many happy days ahead! Just listen to your heart and trust your own wisdom! You're a great mama!