Lovely EVERYONE with a ten week old struggles. It sounds like you are getting out of the house more than I managed when DD was ten weeks old! Although it's not the best idea to be driving a tiny around, I do understand why you are doing it.
The only people you have to worry about keeping happy is you and your DD. Let the rest of them worry about keeping themselves happy.
As for the ex - if he wants to see DD he will make the effort to come to you. If he doesn't then you know he isn't bothered. NOTHING would keep me away from my DD, I'd walk if I had to, and I'm sure you would do the same. If he won't then he doesn't deserve her. You already know he doesn't deserve you. You and DD both deserve so much more than he is willing to give.
She's only ten weeks old - you are doing this on your own, and you ARE managing, you should be so proud of yourself. Your DD will be proud of you.
I grew up knowing my Dad, but I have no memory of him living with us. And I'm glad I don't. My Dad isn't the greatest, track record of putting other things ahead of us, but he's ok. My Mum, though, is THE most amazing person I have ever met. She brought my brother and I up singlehandedly, with very little money and all I remember is a fantastic childhood with a woman who put us first. That's what your DD will have. I don't believe children need two parents, I didn't, I just needed my Mum - and we are so close.
And, this woman who I respect, admire and love so much? She had PND - but I don't remember that - and your DD won't either.
Big un-Mumsnetty hugs for you. You're doing brilliantly, even if you can't see it. And tell that man there is no way you are running around after his ass any more. If he wants to see DD he can drag HIS ass to you.