How to disable payment cart pane when discount brings total to $0
I was able to fix the problem by modifying the "Free Order" payment method's Conditional Action. Free Order is the payment method associated with the Ubercart Free Payments Module and comes pre-loaded with a CA that checks the cart total to see if it is more than $0.01... if it is more than that, it negates that condition and does not allow the Free Order payment method in the cart.

I changed that CA to instead check whether the cart had the discount(s) in question. If the discount(s) was used, the cart pulls in the Free Order payment method, and it all works as expected.