If other email addresses work then either your server is blocking the outbound email or their server is blocking the inbound email. Since the page delays and never sends most likely it's your server blocking it. You'd need to talk with your host and provide them the domain that you're unable to reach.
can I take one of the test invitations and just forward it to the person? Any reason that would cause trouble with their registration?
Invite emails are unique to each invite. They contain the unique invite code for that specific invite email. So no you can not just copy the email over for it to work.