Assuming you're already using the Disc # tag, you can just create a custom tag (Edit > Edit Preferences > Tags (1)) to save the Disc Name part.
(https://i.imgur.com/rDKEFry.png)
Click Define New Tags... and add it there if you want the values saved to the file and not just the MusicBee database
(https://i.imgur.com/hnGkg3u.png)
Then create a virtual tag in the same preferences screen with something like this:
$IsNull(<Disc#>,,"Disc "<Disc#>)$IsNull(<Disc Name>,,$IsNull(<Disc#>,,": ")<Disc Name>)
(Here, <Disc Name> is the name of the custom tag you've created)
Then in the Album and Tracks panel settings, select that virtual tag as the track sub-header.
That will give you this:
(https://i.imgur.com/H1I0yer.png)
1. No value for the disc # or disc name tags. (i.e Shows nothing)
2. Value for disc # tag only.
3. Value for disc # and disc name tags.
This also lets you display a heading for some tracks even if there is no disc number.
For example, "Bonus Tracks" at the end of an album.