I'm inside the zone at the finish — can I call for mark-room on the other tack?

No. Rule 18.1 switches mark-room off between boats on opposite tacks on a beat to windward, so the three-length zone is irrelevant at an upwind mark or finish. Rule 10 decides it instead: the port-tack boat keeps clear of the starboard-tack boat. Ask the tack question before you hail for room.

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