But MH are the ones who charged the fee; they are also the ones who you would have to pursue for (partial) repayment.

It sounds like you should have been charged MH's fee for a checked bag from KUL to SIN only; as I said in my previous post, $120 sounds excessive, but then again I do not know how high their fees are if you arrive at the airport on a ticket with no luggage allowance and haven't availed of any possible advance fee reductions.