The answer is in your question.

1. Convert polygons to lines (features to lines)
2. Merge them with fault line
3. Dissolve output from 2 to single part polylines
4. Select and delete from DISSOLVED lines that share line segment with FAULT line

RESULT:

NOTE solution applicable for topologically correct geometries, i.e. fault lines truly follow boundaries and there are no slivers/overlaps in polygon layer.

Some manual editing possible, consider fault line going up a bit from uppermost point.

The answer is in your question.

1. Convert polygons to lines (features to lines)
2. Merge them with fault line
3. Dissolve output from step 2 into single part polylines
4. Select and delete from DISSOLVED lines that share line segment with FAULT line

RESULT:

NOTE solution applicable for topologically correct geometries, i.e. fault lines truly follow boundaries and there are no slivers/overlaps in polygon layer.

Some manual editing possible, consider fault line going up a bit from uppermost point.