According to that link, though, “Co.” doesn’t actually have legal significance. If there was a “Unichappell Music Ltd.” or “Unichappell Music GMBH” I would agree with your point.
A search on the BMI website for “Unichappell” returns a bunch of entries for “UNICHAPPELL MUSIC INC” (all with the same ID) as well as a “UNICHAPPELL GARVARENTZ” and “UNICHAPPELL PETER PIPER” with different IDs (but no works in their catalogs).
There are only a few works associated with either Unichappell Music or Unichappell Music Co. in MB today, and some of them aren’t even listed in BMI as being published by Unichappell any more, but One Big Sky is listed in MB under Unichappell Music but in BMI under Unichappell Music, Inc.
So if we leave them separate, when do I use Unichappell Music vs Unichappell Music, Inc. ? Only use the Inc. if it’s printed that way on the release?