You should be able to use the iCloud ID in the App Store/iTunes. Another option would be to just stop using the Apple ID and set up Family Sharing between the 2 IDs. That would allow you to keep access to the purchases you have made.
Family Sharing. *
Family Sharing - Set up.
All purchases made using an Apple ID are tied to the ID. Delete the ID and you will need to download them using your new ID. There have been posts by people who have contacted Apple Support and they were told that if you delete an ID associated with an email address, you can never reuse the email address for an Apple ID. Make sure that you sign out of all your devices to avoid issues with apps that use your Apple ID account or iCloud. After your account is deleted, you can't sign out of iCloud or turn off Find My iPhone Activation Lock on your devices. If you forget to sign out, you might not be able to use your device when your account is deleted.
How to delete your Apple ID account. *
Apple ID – How to Delete. -
Delete Apple ID (new). -
What to do after you change your Apple ID or password - Apple Support -