yitayningwut-
First of all, in this case the goy for sure has no neemanus. The worker does not care about any “bad name” that might come out of how he claims he washes the dishes etc..
Second of all, according to CRC, who did moths of research on Starbucks, the Frappuccino base is not certified and has kosher sensitive ingredients. See the link I left above and here:
http://www.crcweb.org/Starbucks%20cRc%20Full%20Article%20April%202011.pdf an extensive report on Starbucks.
As is clear on the CRC’s report, they were not looking for chumros at all. They are also not a “right wing” hisachdus type of organization. Rabbi Fishbein of the CRC is one of the leading kashrus experts in the U.S.A. and heads the A.K.O.- Association of Kashrus Organizations of America. If he has a problem with this drink, so should you.
As far as your Rov, I do not know who he is, nor do I want to know who he his. However, very few Rabonim are experts in the kashrus field. They might be able to answer kashrus questions, but many of them are not up to date with the modern complications which are presented with today’s advanced technology, even in coffee shops.
I am not at all familiar with the Star-K. All i know is that they are far from leaning to chumros. They permit what many kashrus organizations do not.
If it is true, as posted earlier, that they also feel one may not drink a Frappuccino, it might be time for your “Rov” to pick up the phone and speak to some kashrus vaadim who are familiar with the situation.