In Manifold 8? (It's a Manifold 8 file.)
That is easy: use the SQL function IntersectLine(geom1, geom2), where geom2 can be a point.
[Added.] Like this.
OPTIONS CoordSys("Line" AS COMPONENT);
SELECT L.ID, [Branch]
[Line] AS L LEFT JOIN [POINT] AS P
ON Touches(L.ID, P.ID)
SPLIT BY Branches(IntersectLine(L.ID, P.ID)) AS [Branch]
Link a new drawing from that result.
In Manifold 9 there is nothing built-in yet. It would require some more verbose SQL.
That would be fun to write--just say if you need it.