I realize that ebay auto sends them reminders after a few days, but I'm not sure when and how many reminders they receive.   My question - is it better to send them a private message or send them another invoice?

first of all, you can check whether ebay has sent them a reminder - this is in  MY ACCOUNT - MANAGE COMMUNICATIONS WITH BUYERS

see if the ebay default for a reminder has been ticked.

If they have already had a reminder from ebay, I would just send them a polite message .....this is one I use supplied by these boards members some time ago...

Hi: ???

I have just been checking my payments received and see that I have not received a payment ???  from you yet, as it is now ? days could you please check your records to see if the payment has bounced back to you for some reason, or email me within 24 hours  if there is a problem.

Then if they don't answer after 24 hours  I start the unpaid item process

(resolve a problem box)