Yes, we're on the same page then.  I'd just put it slightly differently - pay tax to MN (as they have the primary right to tax that income) and claim a credit on WI return.

In your original post, you said the diagnostic says you must have wages for the withholding.  Your latest post now says the error tells you "MN requires a 7 digit State ID" and that is done.  What exactly does the diagnostic say and what is "done"?

Your latest post also says "MN income amount when there is a withholding amount".  Question is: Did you code the unemployment income to MN or WI?  You need to make sure it is coded to MN.