Bundle Builder was designed to primarily integrate with the cart page. Bundle Builder can integrate with your theme's drawer or Ajax Cart. However, when a customer creates a bundle and proceeds to order it, they should always be redirected to your cart page by default - even if the theme is set to use the cart drawer.
Custom integration
In order to integrate some third-party or customized cart drawers with the Bundle Builder app, customization may be necessary. Code changes for this process will include edits to javascript and liquid code already present, and it's recommended to seek a specialist or developer for these tasks.
If you're not confident with changing code yourself, we'd recommend finding a Shopify Expert to work with. HeyCarson and StoreTasker are recommended partners of Bundle Builder - both specializing in website development and offering free quotes for their services.