The pet fee is the least of your concerns.
So what should you require IF you wanted to continue to do business with them?
Rental: Minimum of $4,600. That would be first, last, and two additional months. Some states have limits on what can be charged for a security deposit. So you could just make that a 1-month security deposit but charge, first, last, and next-to-last. After all, many landlords charge first, last, and security deposit to tenants with GOOD credit. These folks don't have that.
Lease-option: You don't say what the purchase price of the home would be. You also don't say what rent credits, if any, you'd apply. Those are important to know. At a minimum, 3% of the sales price or $6,000, whichever is higher. Break the numbers down this way: $1,150 for first month's rent. $100 security deposit. The rest is the non-refundable option fee. That way, if you have to kick them out (or even if everything goes well but they decide not to exercise their option), you get to keep the bulk of what they've given you.
And have a lawyer review all the documents.
Hope that helps.
The fact that you are questioning commiting to them in itself should pave the way for your final decision. Because they appear to be a nice couple won't guarantee you will see your monthly rent.
Good luck with your decision.
All the best
Dave & Lisa