I eventually figured it out. I was trying to remotely fix this, not having a direct access to the printer and not seeing what it printed and how. But eventually I got access to a printer and then it didn't take long to fix.
It seems that there were some defaults or setting left over from the previous print jobs that cause the problem. I added some resetting of values in front of the printing and after that it properly printed out just one label.
And yes, the ^FS was not needed at the end of the comment with ^FX.
Here is the code that printed out the same thing correctly (there might be some unnecessary commands but this one worked so I left it at that):
^XA~TA000~JSN^MNW^PON^PMN^LH0,000~SD15^XZ
^XA^JUS^XZ
^XA
^MMT
^PW0355
^LL0305
^FX Test for long barcode
^BY1,2,50
^FO20,10^BC^FD>6UIQ-2013P-MR-BU^FS
^XZ