You can still pay with gift cards even if you've set up auto-pay. When you're ahead in payment, deductions wouldn't be made from your bank account that month (pay at least a week before the 15th).
Paying with gift cards sounds like a great way to save 5-10% with minimal effort. I'd definitely do it if we lived in the States and had one of the Target/Sam's/BJs memberships! Please make sure you buy Disney gift cards (good for most things Disney), not
Disney Store gift cards which can only be used at Disney stores for merchandise.