As I indicated before, I don't see any way around this provision. It refers to "any animal, including household pets" and then excludes only one dog, one cat or two cats and two dogs. Otherwise it requires written permission in advance. If you had that, you wouldn't be writing this.
This is what you get when you move into a subdivision. If you want chickens, goats, etc., you need to move into the country or be in a city that permits them but not be in a subdivison with its own restrictions. People set up those restricitions because they specifically don't want to be around certain things.
If you really have your heart set on living in that subdivision, you might try running for the association board the next time they have an election. Usually nobody knows about those things so if you can get a group of people you know to show up and vote, you have a good shot. Better yet, run an entire slate and pack the meeting. That way you will have controlling votes once elected.
Cities are different. They may have diverse populations and diverse housing stock and neighborhoods. When Chicago tried to ban chickens a few years ago, the Hispanic community showed up in force and killed it.