Hi Emily,
I haven't read the other responses. Just going to go with my reaction to your post. I know you asked for advice on communicating with a depressed co-parent, but it sounds like you've tried talking with him, and he just persists in his self-pity, basically saying, "I'm going to do/say whatever I want to regardless of the effect on my daughter."
Your ex is being utterly inappropriate with your daughter, and that is putting it mildly. He seems unwilling to see her as the toddler she is and is completely putting his needs above hers to her detriment. This is harmful to her now, to her developing sense of self and how she views relationships, and to the woman she will someday become.
He is the one who should be caring for her when she is with him. He is supposed to be the adult because he is the adult. Yet, he is trying to reverse the roles. He is setting her up to be HIS care-taker. It's all about his feelings, his depression, his sadness, his tears, his regret about not being with you. It's all him, all the time.
Where do her needs fit in with him? Her need to feel loved unconditionally by dad, to feel secure, to feel like dad's in charge and will take care of her. At nearly three, her biggest worry in life should revolve around monsters under the bed, not worrying about whether daddy will be okay.
Yes, it's okay for very young children to see parents sad ONCE IN A WHILE. It's part of life and that's how kids learn about feelings. But when dad is in a constant state of depression and starts using his daughter, who's still under three, as his sounding board, then there are red flags waving all over.
In addition to his constant sadness, he is discussing ADULT business with your daughter. Whatever drove you two adults apart should not be of any concern to your little girl. She should only feel that mommy and daddy both love her and will always be there for her no matter what. He should not be lamenting the fact that you're no longer a couple in front of her (that's what therapy is for). He should not be pointing blame at your new partner (sounds like that's what he's saying to her).
Based on what you've said, it doesn't sound like a meeting with his family will be helpful. What you have said, however, is that his seeking treatment was part of your legal agreement. So, my question to you is why have you not pursued this? If you've got legal ground to stand on, take that stand and protect your daughter.
Your ex's behavior, left unchecked, has the potential to do real harm to your daughter. (Read up on children who are raised by chronically depressed parents if you need more convincing).
If it were me, I'd contact my attorney, go back to the judge and get some teeth behind this order. He wants visits? He has to be stable, appropriate, and have his mental health issues under control by working with a qualified mental health professional.
If it were me, I would also want supervised visits until such time that he can demonstrate proper parental behavior to the court.
Sorry there is no magic answer to make him communicate with you. I truly hope he gets the help he so desperately needs.
I wish all of you the best.
J. F.