Can we stop a customer's billing moving forward without canceling their subscription?

Yes, there are several ways to handle this specific use case that we could recommend. The first would be to simply edit this customer's individual subscription and change the price to $0. With that change being made, even if the customer decides to remove their stored billing information, the subscription will remain active as all future invoices will automatically be generated with a $0 due amount and close.

Another way to handle this would be through the use of a coupon. If you create a new coupon and have the discount type set to percentage and 100%, you can then apply the coupon to the customer's account/subscription and have it begin discounting their new invoices moving forward. Coupons are good if you only want to temporarily stop the billings from occurring with this customer's subscription.