Hi,
How about automating some of it with a mass action.
For example, you could create a mass action with a trigger "before an order is updated", a filter on the column "order_status" with the value being equal to the namekey of the "Awaiting Payment" order status (you can find the namekey when you edit the order status in the System>Order statuses menu), and with two action "update the values" :
- one on the column "order_payment_id" with the type "int" and the value being the id of the Stripe payment method in System>Payment methods
- one on the column "order_payment_method" with the type "string" and the value being "stripe"
That way, you just have to change the order status and the payment mehtod will be changed automatically for you so that the link will be in the email notification.