Hi,
I'm not sure I understand your problem.
The fact that the selected payment method changed doesn't necessarily lead to the cart view being refreshed. And similarly, when the cart changed, it doesn't necessarily lead to the payment method selection area being refreshed.
A refresh of a view on the checkout only happens when the system detected that the change made by the user can or will lead to a change in another view.
So for example, if your payment methods have different "price" setup, when you change the payment method selected, the cart view will have to refresh since the payment fee in the cart will change. But that's not the case on your checkout.
Similarly, I did some tests on your checkout but I don't see anything changing in the payment methods available when I change the cart after a full refresh, nor do a see anything changing in the cart when changing the selected payment after a full refresh, which menas that there is no reason for the cart / payment views to refresh themselves.