Seems like there’s a few different issues:
- is H taking S to his place in the mornings (versus taking S out to breakfast or washing and putting away breakfast dishes?). Switching to afternoons will not keep him from taking S to his place. You simply need the lawyer to fight to have him reveal his residence. A nanny cam could tell you what’s happening with breakfast. It could also reassure you that he’s behaving appropriately with your toddler. (But look into your local laws, if it’s not admissible in court do not mention your findings to him. It could be illegal where you live).

- H taking all afternoons would eliminate the friction of waiting for him in the morning. Is there a benefit for him too that you can use to sell it? Like “H, I know you’re really not a morning person, how would you like to do two afternoons instead? “

- I agree with trying a “softer” approach first. The more you can paint things as a “win-win”,