Not sure if you are still reading the responses but I want to respond anyway.
But H really wants to try and fix things and is willing to give whatever assurances I need e.g. all finances in my name, passwords to all of his accounts, phone tracker etc
He wants to do this now, because he assumes the lack of trust might last a few months. I bet he doesn't envisage that the lack of trust and feelings of resentment and betrayal might last for years, maybe decades. Has he really thought that one through? That these changes could last years?
What he's proposing is introducing an unhealthy dynamic to try and patch things up. How long exactly is he willing to keep this going. He might be all for it now because he's desperate, but as he feels things are sinking back into something that resembles normal, he might not like what he suggested any mire.
Even if you both did all these things, how would you know all of your finances are truly in your name? Wouldn't he just be able to open a small account elsewhere? Get another phone? Create new email addresses for clandestine uses?
I think you need to ask yourself exactly where the bar is? How low does he have to go before you split up with him? I bet that initially, you probably would have said this exact situation, but now you might be trying to decide if that really is a dealbreaker for you. I sympathise, because breaking up with a long term partner is a huge thing to get your head around, practically and emotionally, and you have the added pressure of a child in this as well.
I would go down the following list, and ask yourself if you are willing to forgive the fact -
1)That he cheated on you
2)That he exposed you to STDs by going outside of the relationship sexually, and with women who might have multiple 'customers' a day. Condoms don't prevent all STDs (depends where the sexual fluids end up going), and that's giving him the benefit of the doubt that he did use condoms, and that none of them broke. Not only did he expose you to possible STDs, but he exposed your unborn baby to them as well if you were sleeping together whilst pregnant.
- That he slept with prostitutes, with all of the ethical issues that brings with it.
4)That he slept with them for years - this is no one-off. He is a regular customer, and I wouldn't be surprised if he did this before you met. He's just carried on with the habit when he felt he needed to.
- That he most probably would have continued doing all of the above if you hadn't found out.