Need to pair a CableCard

... How can I pair my card online?

They don't provide a way to do that. They claim to, but as you have seen, it doesn't work, and hasn't for years. You'll have to call again, or chat. I've had success (and failure) using both methods. Expect much confusion, ignorance, and resistance.

ETA: Also see https://tivoidp.tivo.com/tivoCommunitySupport/s/article/CableCARD-Troubleshooting and the post following this one.

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I recently upgraded my router. Ever since I did, my TiVo stopped showing live TV. I am assuming the CableCard needs to be paired to the account again. ...

It would be odd for a router change to require the CableCard to be paired again. It's more likely there is a signal problem. Are you seeing any error messages or codes? Also see "How to check the Signal Strength and Signal Lock" about halfway down https://tivoidp.tivo.com/tivoCommunitySupport/s/article/Troubleshooting-Digital-TV-Signals.

Please be aware that there are 2 kinds of responses in this Forum: Replies and Comments. When you Comment on a post by scrolling down to "Comment on this post here...", I am notified of your response. But if you select Reply, I am NOT notified and may not be aware of your response.

I did end up calling again on a Saturday morning, and Miracle of Miracles, got through to somebody who knew what they were doing; the card was paired within 10 minutes.