*She might not be putting capital into the house but she is servicing the loan and the OP will benefit from the capital appreciation. She's doing him a massive favour.
He's not getting any benefits because he's not the child's primary carer. As he guarantees to have his child for two nights a week, he only pays 5/7 of the full maintenance that would otherwise be due.
Why would the parent who can only guarantee to have his child for two nights out of seven be the one to decide of the child's school lunch arrangements? Has the OP offered to drop the lunch off at school five days a week, if he feels so strongly? However, this will have nothing to do with the child's lunch, it will be about tightness or control.*
She's not doing him a massive favour at all. She's living in their house, he's having to rent at a bigger cost to him.
He has the child 3 nights, but pays more maintenance than he legally has to.
He's not the child's primary carer, but three nights is significant. He isn't getting 3/7 of the benefits is he? No.
He wants to decide the lunch arrangements because the mother doesn't want to pay, and the child doesn't want to eat it anyway. It makes more sense for everyone imo for the child to have pack lunches. I really don't understand why the mother won't just let op do it as he has offered, she wouldn't be paying then would she? No.
It is about control, on the mothers part. Op wants to. Hangs to pack lunches because he thinks the child will eat more, which is a sensible reason imo.