Everytime I'm trying to post to the 'cat the system is disturbed by unaccessibility ... Tried for about 5 hours. Let's hope it works now:

The metre of 10/8 is very, very unusual in German folk music. Did you hear right?
The Kat could be Katz, dialect for Katze = cat.
There is a song "Unsre Katz hat Junge" = our cat has kittens (we sang it as young scouts), but the line you mentioned is not in this song.

But Kat could also be part of an abbreviation for the name Katharina = Katherine, Kate. The rest of the line is Bavarian, where the name is Kati or Kathi.
Translation is "our dear Kate will finally get one, too." One here means a lover, a man, a husband. I think this is the right path to seek further.
Not found in ingeb.org nor with Google.

Sorry
Wilfried