No, it's not your husband's responsibility to convince you against your will and what he should be saying is, 'Look Mum, we love you and are very grateful for all your past help but things are different now. You know the door is open, why not visit?' The answer is, of course, because she's in a snit 
Yes, she's lonely for your darling son, her grandson but you have not barred her from seeing him, or even limited her. The circumstance changed, that's all and she has to accept it.
Your husband should be showing a loyal, united front, not asking his Mum to ring you as if it is your fault!
When my first son was born, a prem baby needing his mother's love and care, I had no alternative but to leave him with mil and work in the business my husband had just purchased.
I longed to be with my baby but could not. So I know how you feel.
I was away from the farm we then shared with my parents in law (who were lovely) from 7am, to well after midnight. It was a gruelling job with a long commute which eventually broke my health.
So please stick with your decision; these days never come again, even if you have many more children. This one is unique.
