Back to the twelve year old - I'm kinda bummed she's involving him in your relationship. I feel like kids shouldn't have to be involved in their parent's dates, until it is really serious and heading towards marriage.
That being said, it can be a very expensive proposition to hire a sitter all the time. It costs my friend about 50 bucks to go out for the evening with me, hiring a sitter for her 9 year old. Two dates a week would add up to $400 a month - more than most single moms could afford.
Soooooo---what's the solution? Seems like most people in this position date when their ex has the kids - but I'm guessing this woman doesn't have an ex who takes the kid? You could offer to help pay for the sitter. You also need to realize, this kid IS her priority and she's a package deal.