Do NOT ever sneak pets into a renting unit, no matter the species. Your landlord WILL find out and you WILL be out on the street in a week.
Unfortunately you need to live by the rules your landlord sets, it is their property and they have the right to ban animals if they choose. Check into buying a house if you have the credit for it. It's a big step but there are 0 down USDA loans available and it's often better in the long run. Good luck!