Will the bundles keep inventory sync?
Yes, when an order is paid, the individual items are added as free items to the order. The inventory of the individual items will be deducted. So the customer will see the individual SKUs appearing on the order within ~10s.
However, if the customer adds a bundle to cart but doesn't check out for a few days and one of the items is sold out, it may lead to an overselling issue.
BYOB relies on Order Edit to keep inventory sync. However, there are some limitations to prevent order edits from proceeding. Please refer to this article for more info: Why aren't my orders edited?