With the help of Tom (K8TB - thank you again) it works now..
I duplicated his serial settings ("Mode 3" char-timeout and only 1 stop-bit not 2 as Elecraft is suggesting for the K2) and it worked using both HRD and TRX-Manager.
So the reason for the faulty transfer was my using of 2 stop-bits.. That's why TRX-Manager and HRD were working directly (because there I cannot set stop-bits) but not via the Remoterig boxes.. Simple reason, big headache.. :-)
Does anybody know if the char-timeout mode is recommendable for remoting via the internet (traffic wise)??
Thanks again for your help Tom, it was necessary for me to see a setup and the settings which are working to duplicate this.. The manual is a bit counter intuitive because I was tempted to use "Mode-4" first (it is saying Elecraft)..
I'm looking forward to do the actual remote setup using the K3 in some weeks (it's more then 400km away from me).
Best 73 de Mario, DL3LSM