This doesn't sound like anything I've felt, but I think you should definitely see a doctor. They won't ask you to strip down! And if they do, it's well within your rights to refuse. Trust me, I know.
As with any pain, if it suddenly gets too bad, a trip to the ER is always a good thing. Maybe have an ultrasound done too, to see if they can see anything. You don't need a referral or anything to do that. Just make an appointment.
I have and I felt is just as randomly as you, in fact I've only ever felt in twice in my entire life with a three year gap between each one. Like you, I was on my period too.
Every time I stood up straight I felt as if my ovaries were being stretched and pulled, like I was pulling on something that had been tightened.
I don't know what it was, but I put it down to cramps as it was my period. I haven't felt it in 6 years now.
As with any pain, if it suddenly gets too bad, a trip to the ER is always a good thing. Maybe have an ultrasound done too, to see if they can see anything. You don't need a referral or anything to do that. Just make an appointment.
Do keep us updated!