I would just save the email that the invite was sent to and save the invite against it. When the user signs up with that email I would know they have an invite

I would just save the email that the invite was sent to and save the invite against it. When the user signs up with that email I would know they have an invite waiting for them. There's a chance they might not use the same email address to sign up but this makes a much cleaner flow for the user.