Tired on the weekends? Well, that's a great thing to say and all but when you have kids you sort of just get past that whole "I'm too tired" thing. My only request is when I get home from work that I have 10 minutes or so to change clothes and "Take care of me" After that, I'm on the team for the rest of the night. On the weekends, there are tons of things to do that don't involve heavy physical exertion. What exactly is physically exerting about laying/sitting in the floor and showing your daughter blocks, or pictures in a book. As was said, peek-a-book always works. One of my favorites was a dish towel or cloth napkin.. Just laying it over their head and pulling it away, BOO! It pretty much never fails at this age. (after the comfort level is there of course)
And the whole thing about her hating him. With my wife in school and me being involved so much, I put myself in "uncomfortable" situations a lot. Even exclusively BF babies can have a mom break. Once a baby has eaten, it's usually 2 hours before they have real desires for it again. Why not say "Here" and take a walk around the block? At some point, dad has to deal with the uncomfortable situation.
My daughter now loves spending time with me. It took some uncomfortable times for her but it's made us such a stronger family. What does mom do that dad could do? Do you show her flash cards? Pictures? While you're doing one of your activities one time, have dad sit down next to you all and join in the process. The moment she continues interest in the activity with him, get up and go do something else. Even if it's only a few feet away. We still do this now as a way to transition between people doing activities with our daughter.
There is nothing inhumane or insensitive about leaving a child in the care of a caring, loving parent. Even if they protest some. I would say that the separation anxiety decreases, but it doesn't. And it normally shifts back and forth between mom and dad. With our daughter being 16 months old, she frequently has to be heavily distracted so I can leave a door without her wanting to follow along.
The attitude of, "I just won't try" doesn't work. If you both want him to be involved and have a great relationship with your daughter, then they have to continually be placed into situations where they can nurture that success.