I have 2 separate Apple IDs - one for personal and personal devices. Then I have another separate Apple ID and iPad device for the business that other employees can access as well. How can I log in to ONLY the Apple Music app on the iPad to use in the store?
The Apple Music component of iCloud is tied in to the main AppleID signed into the device. So firstly sign in to the device in the Settings app with the AppleID that you want to use the for Apple Music.
Still in Settings, go to the iCloud section and turn off all iCloud services that you dont need e.g. Photos, Mail etc.
You will probably want to increase the security so no changes can be made to the account without a passcode.
Then go to the App Store app, tap Featured and scroll to the bottom. Tap on AppleID, then Sign Out.
Then Sign In with the business AppleID and you will be able to install all the apps associated with that AppleID.
If this is inappropriate then it sounds like playing Apple Music directly off the device may not be a good solution. There are many options for playing back music from a local server e.g. the Synology NAS allows using a free app DSMusic for playing music from a library stored on the NAS.