We're private too, @Louise C15 and this was a really bad experience for us when we launched (only been live since August 14) and our community came to Tribe from a private Facebook group and was monetised (not anymore, long story).
Because the efficiency and reliability of the email invites was patchy at best, I ended up running a pipeline in Pipedrive to keep track of it.
After researching on here and asking Support for help, I had a go at setting up the Send Grid app but because I was time poor I didn't get a chance to put much time in it. The Tribe guys sent me some info but I'm only a volunteer for a charity and don't have access to a developer etc.
I did ask Send Grid for help but they never got back to me.
I was also going to try using Zapier but again, ran out of time and patience when I had new members complaining they couldn't get access.
You can't resend private invites as far as I know. You would have to revoke them and send them again.
I hope this helps, yell out if there's anything else I can help with.