Whoever does the inviting, pays. If the woman wants to argue about it, tell her that’s your rule and that she’s welcome to ask you out and pay then. If she still wants to argue, then you know that she’s more interested in being a feminist than in being a gracious date.
Argue about it? That’s about when you say “enjoy the cats, and lonliness...”
“gracious date.”
There is the key term. Manners are no longer common.