There seems to be a confusion between Apple Pay, and Apple's media stores payment mechanism.
Apple's iTunes and App Stores do not use Apple Pay in any way shape or form.
If you subscribe to Disney+ through Apple / your Apple account, then Apple Pay is not involved in any way.
You can set the default payment method for your Apple ID to be used for recurring subscriptions separate to that of the default card used for Apple Pay because they are 2 different things.
To modify the payment method used for subscriptions charged by Apple, through your Apple account (again, completely unrelated to Apple Pay) follow the steps in the support document linked below.
click here ➜ Add a payment method to your Apple ID - Apple Support
The payment methods added to your Apple ID are charged in the order they appear in. 
You can tap on edit and then drag the payment methods there up or down. The top payment method will then be used first for the store payments and recurring subscriptions.
The Apple ID balance cannot be moved from the first position and will always be the first thing Apple attempts to charge for iTunes and App Store purchases and recurring subscriptions, followed by your other payment methods in the order they appear in.