Without seeing the setup it's hard to tell if this isn't just a configuration issue with the app.
Regardless, there are different apps specially for syncing inventory that can be used to support this use case.
Also automation apps like usemechanic can be scripted to run custom business logic such as sync a bundle.
However, the issue isn't with tracking the inventory. It is with the increase with price when the product is added to cart.
You specially mentioned it as a condition, for the price increase your first step is to contact the APPS support to see if your theme needs to be modified to display the dynamic prices, or if the theme has problems.