Will Shopify automatic discounts work together with bundle discounts?
If the customer qualifies for the bundle discount, then the automatic discounts won't get applied. This is a limitation posed by Shopify, as draft orders are not affected by automatic discounts.
However, if the customer doesn't qualify for the discount (e.g. he doesn't have all the products from the bundle in the cart) then he will go to a normal checkout and the automatic discounts will get applied.

If you want to use a sitewide discount together with bundle discounts, then the recommended solution here is to create a "volume discount" in Bundler - Product Bundles app for all products, which can represent your storewide discount. This will then make Bundler apply it's own storewide discount.