Your DH needs to get a grip and stop the constant checking and let her sleep. He is doing neither of you any favours by letting his anxiety get the better of him.
She needs to be allowed to establish a habit of natural sleep patterns, hopefully gradually extending her nighttime sleep periods. This won't happen if he keeps on disturbing her.
If you have a double bed to yourself, take the baby in with you for the night.
The DH needs to be responsible for completely taking over some housework - planning and execution, not just doing what you tell him after you have done all the planning.
Sometimes a baby will accept a dad's company better if he takes her away from the usual environment for the times he is with her. He could take her out for a walk somewhere, even around the garden, where there might be interesting distractions for her.
(milk yield as someone said only shows what you can express, not what the baby is getting, and comments about richness of milk, quantity, etc are really not helpful unless failure to thrive is at issue)