IINM the tatfus possibilities at Starbucks include basar bechalav, so...
I think it's unlikely, as they don't serve meat from behind the counter.
It's usually all prepackaged sandwiches and salads which probably come from out of store, but even if not, that drop of meat on the knife is nothing compared to the tons of dairy equipment they use every day. Also if It's pork and milk there's no problem of Chanan (acc to mechaber). More things to clarify is if the water that washes the dishes is yad soledes, if there's soap involved, if there's no shishim against the treif products and etc. My guess is that for Sephardim the biggest issue is treif being served, not so much a problem with the Keilim.
One of the things the agencies say is assur is plain coffee from the machines because it touches glass Keilim which were assur'd from Chanan. From a Sephardic perspective this chasash doesn't even start.