Yes. A short update is professional.

The referrer invested time in helping you, so closing the loop shows respect even when the result is disappointing.

You can thank them again and say that you were not selected to interview this cycle.

Keep it brief, do not ask them to challenge the decision, and maintain the relationship if you may reapply. If useful, this guide on consulting referrals gives more detail.

I'd definitely update them. Short thank you + outcome is enough.