Fire door modification beyond trim allowance
How likely is it that this would be signed off by Building Control?

Highly likely. The way you describe the carpenter's modification proposal it would likely not be easily detected.

If it was signed off, in the event of a fire, would we be covered by our insurance?

If they knew about your modification, no. However; unless the door in question contributed to the fire spread there would not likely be a close forensic analysis of the door, right?

If we rented the house, in the event of a fire, would we be liable?

You would be liable if the door failed to prevent fire spread as designed due to your modifications. Held accountable? See above.

If we sold the house, in the event of a fire, would we be liable?

You would be liable if the door failed to prevent fire spread as designed due to your modifications. Held accountable? See above.
I am not giving legal advice, and you should not trust legal advice from the internet. Deliberately violating building/fire code to make your job more convenient is not the right thing to do. If the worst happened, you and the carpenter and the inspector and the door manufacturer could be in court swearing oaths of truth. How would you feel if your renter's children were killed because the modified door failed to prevent fire spread?