I have questions to ask - firstly, I don't have the full details of why you have separated, my response applies if there was no physical/mental violence or neglect from the father of your child.
It appears that your ex is in love with his child and wants to play a part in his life in a consistent way. How fortunate for your child to have that from his father in spite of your relationship not working. Your ex is showing extreme maturity in wanting to mediate, especially if that is the only way you both can talk without arguing. If you are not willing to mediate, then of course his next step would be court - he is a desperate father who's needs are not being acknowledged in his role as a parent. I understand you feel worried and anxious - imagine how he feels every day he wakes up without seeing his baby's gorgeous smile or how he feels after working a hard day and not getting to spend time winding down with his gorgeous baby.
I understand your ex wanting to show off his baby to his relatives, but this is probably not in the best interests of bubs, and I'm sure this is able to be communicated to your ex if his needs are acknowledged in consideration with yours and bub's. You seem rushing to seek a lawyer, I would urge you to seek mediation as your husband suggests and give him gthe recognition he deserves in suggesting this. I hope you can find some peace in your life by accepting the father of your bub - he will never be denied this title. Good luck