You may have to negotiate a little bit. Surely one Sunday for you, one for her sounds reasonable, as is the 'no religious responsibilities before 18' bit. But from her perspective she is giving up big things: 2 Sundays a month, the spiritual well-being of her children, and if she had pre-18 priesthood in mind for her children, that as well.

Perhaps you could start with no spiritual responsibilities before 18, and either one Sunday for you plus a Saturday with wife and kids together in the library, watch a movie etc. Who knows, you might be able to coax her out of Mormonism a little bit on those Saturdays.

Let her get used to it, see it's not the end of the world. The kids might even start asking for more 'daddy-days' themselves.

Hope that helps.