So this needs to be done online and not in the store?
Yes, in order to get the Red Card 5% additional discount, you must use the Target gift cards to buy the Disney gift cards at target.com. Have your default payment method at target.com set to Red Card, but then at the last screen when checking out, you are actually able to use the Target gift cards to pay instead of your Red Card. (At Cardpool or Raise, most often you are buying electronic Target gift cards, which are e-mailed to you within minutes in my experience.) Obviously there may be some small balance left over if your Target gift card amount doesn't exactly match the Disney gift card purchase amount. That can be left to go on your Red Card.
Cardpool and Raise don't always have Target gift cards, and when they do, some times the discount is only a percent or two. Right now during prime Christmas shopping season, Target cards seem to be going very quickly and are at pretty small discounts. But you just have to set an alert and/or keep an eye out. To beat the original 12% discount in the OP, obviously you need to wait for when you can get a good Target gift card discount.
So I think this is the summary!
Catch a good Target gift card discount on Cardpool or Raise or one of the other discounting sites and then:
1) Ebates or Topcashback to link to Cardpool or Raise
2) Purchase discounted Target gift cards using your cashback/miles/whatever credit card
(so you now have some cashback from Ebates/Topcashback and some credit card rewards and Target gift cards for which you paid a percentage less than face value)
3) Ebates or Topcashback to link to Target.com and put Disney gift cards in your cart
4) Make sure your account settings state your Target red card as your default payment method
5) Checkout at Target.com but change to pay with Target gift cards and use your discounted Target gift cards
(so now you also have cashback again from Ebates/Topcashback for the Target purchase, you paid using the discounted Target cards and also got 5% off the price of the Disney gift cards)