I love Jonathan Coulton and this is an interesting question. However, I think this is just regular Coulton humour.
The family is obviously forced by the robots to stay and work on the asteroid, that's what the lyrics are all about:
Merry Christmas from Chiron Beta Prime,
Where we're working in a mine for our robot overlords.
Did I say overlords? I meant protectors.
Merry Christmas from Chiron Beta Prime.
The robots are apparently redacting the message preventing the family to send out a "hidden" emergency message:
That's all the family news that we're allowed to talk about.
We really hope you'll come and visit us soon.
I mean we're literally begging you to visit us.
And make it quick before they [MESSAGE REDACTED].
The above lyrics snippets are from the official lyrics site, and I highly doubt that there are any "hidden" lyrics behind it. After all, MESSAGE REDACTED is part of the lyrics.