I'm not sure about ultrasonic soldering. Whats the temp the heating element reaches? If its realtively high you may be better off using a purely mechanical connection. Ive never heard of SiC wire. Is this the same as a SiC fibre used for strengthening in metal matrix composites?
I'm just guessing that the SiC is the heating element and the Cu wire just gets the current there. you may want to insulate the Cu wire from the SiC (heat not electric), maybe using the mechanical connection. that way the Cu wire won't wick all your heat away.
Nick