SQ is not going to refund anything if this is a LH plated ticket. SQ will not get paid by LH for a flight where pax that has been rebooked to LX so there is no nothing to refund. Any refund will have to come from LH/whoever issued the ticket and collected the payment from the pax.

If SQ issued the ticket, then SQ is the obvious party to request a partial refund from.