Technically, discount codes can be combined inside checkout pages.
You can toggle this option inside the app's advanced tab:
However, Shopify does not support all combinations, as detailed in their guide:
To overcome that - we developed a flexible ability to support discount stacking prior to checkout.
If you want to allow using extra discount codes on top of your bundle discounts, follow these steps:
Open the app > open the advanced tab > turn on Additional discount codes:
Once you do, an additional input box will be added inside the cart page:
That is, if your cart is showing the cart discounts as expected.
If not, and you can't see the discounts at all - follow this guide.
(1) The extra discount box will only show when bundle discounts are applied.
(2) Customers will need to enter the manual code in the input box to activate the code.
(3) Usage limits and free shipping for selected countries will not apply when using discount codes through Bundle Bear.