Should I ask the instructor for a make-up exam for attending a PhD interview at another university (and risk my standing)?
You don't have to tell your instructor why you want to reschedule your exam.  Just tell her you have a conflict with a personal matter and ask if it is possible to take the exam on a different date.

She might say no, but if she does, then she would probably have said no even if you had told her the real reason.  Usually exam makeup policies are either "flexible" or "no makeups except for medical reasons or conflict with another course".  Attending an admissions event at another university isn't any more compelling as an excuse than "generic personal business".

If she does say no, then you're going to have to choose between the two.  It probably makes the most sense to attend the interview and then come back for your exam, skipping the second day of the event.  It's probably not wise to skip the exam - you don't want to get bad grades if you are looking to apply elsewhere.