Try setting From Domain to the same value you have in SIP Server, and From User to the same value you have in Username.
If no luck, look at the format of the number sent in the To header on an incoming call, and use that format for the destination number on your outgoing call. For example, 4176387XXXX or +4176387XXXX.
If still no luck, check Swisscom documentation for other header requirements (P-Asserted-Identity, Diversion, etc.)