Your h is being wholly unreasonable and if he "can't cope" with your ex being dc's dad he shouldn't have become the dc's stepfather.
You share parental responsibility for your dc with your ex and it's imperative that he should know the name, address, and phone number of his dc's GP in the event of an emergency and, similarly, he should have your/dc's address.
As for communication between you and your ex being conducted through an aunt, how ludicrous is this? You are two adults with a shared interest in promoting the welfare and wellbeing of your dc and, regardless of what went on when you first took up with your h, are no doubt capable of communicating direct with each other in a civil and courteous manner.
I'm sorry to say that your h sounds a deeply unpleasant and controlling piece of work and, if he's not prepared to allow you to exchange information direct with your ex, I suggest you ask him how he'd feel if you divorced him for his unreasonable behaviour and he found himself not knowing his dc's address or details of their GP and had to conduct any communication with you through a third party?
With regard to your h verbally abusing your ex whenever he sees him, I have no doubt that the police will advise him to desist should your ex make a complaint - which I sincerely hope he will as someone needs to take your self-entitled h down a peg or two.
In answer to your question "what am I not doing right?", what you're doing wrong is allowing your h to dictate terms when the right thing for him to do is to butt out and leave you to freely communicate with your ex as and when it's necessary for you to do so. .
As I find it hard believe that your dc is unaffected by this highly unsatisfactory situation I hope you'll take the necessary steps to remedy it asap.
Should you need further confirmation that your h being unreasonable I suggest you make contact with your nearest Women's Aid service www.womensaid.org.uk/domestic-abuse-directory/ as I also find it hard to believe that he confines his controlling ways to this one issue.