Pap smear on period?

So I have a physical checkup pretty soon at my local clinic, but the issue is my period is unexpectedly later than unusual (like 10 days late). Meaning I may be getting a pap smear on my period which I heard that you're not supposed to do.

The problem is I'm unable to reschedule, because the next available appointment is 2 months out, and I will be leaving for grad school before then. I feel really guilty about this because I'm overdue for my first pap smear after putting it off for a few years and I feel guilty about cancelling and potentially making the people at the clinic more mad than they probably already are at me.

Don't know what to do. Any advice?