Is a customer able to add an additional discount code to the bundle offer at the checkout point?

If you want a very quick answer, "No".

Explanation:

When customers add a bundle to their cart, we assign a discount code to that offer. Shopify has a policy on most of its plans that it doesn't let multiple discount codes be added to a single cart. So when people reach the checkout point, they will not be able to add a separate discount code to the bundle.