Answer:
I don't see any difference between claiming illness or claiming a girlfriend minds. However neither is fully honest and additionally can cause a Chillul Hashem if/when it slips out that you were not being honest.
Generally I find the best solution is to be honest and to say that my religion does not permit me to have physical contact with an unrelated female. Most non-Jews will understand and respect you for standing up for your principles and beliefs, although not-yet-religious Jews may be likely to object.
Saying "I wouldn't mind doing an aveira" is not somewhere you would want to go.
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