I am not a UI designer, but I guess that an "Invite Friends" or "Gift Link" should be in a place that is visible, for example, top left, like "Feedback Help Invite Settings". Not sure if you guys A/B test, but that would be a good use case for it.
As for the bummer, it depends - Gmail had a total (not monthly) quota of 50 invites back in 2004/2005. Sounds sufficiently high to me. But you could also simply increase the magic link quota again as soon as somebody signs up.