Go with yor first instinct. A High German umlaut over an 'A' will be pronounced like the 'a' in Mädchen.
There are, however, many many german dialects, and it could be that in certain regions (Austria, Bavaria especially) that this is pronounced differently.
But in High, correct dictionary German the name Käthe would be pronounced KAY-teh, like you thought.
A beautiful and rare name amidst all the Tims, Toms, and Emilys that are all the 'rage' here in Germany.
There are, however, many many german dialects, and it could be that in certain regions (Austria, Bavaria especially) that this is pronounced differently.
But in High, correct dictionary German the name Käthe would be pronounced KAY-teh, like you thought.
A beautiful and rare name amidst all the Tims, Toms, and Emilys that are all the 'rage' here in Germany.