Suggest you escalate to the floor supervisor immediately. If no joy, then escalate to the manager who sits far away in the airport office. Usually staff refuse to honor policy because they don't know how to do it and figure if they rebuff you enough, you'll just eat it. This requires incredible patience. Also, do NOT let them move you aside so they can serve other passengers, until the manager shows up. They'll try to move you to other counters...just stand still. Two people can play the game of being a rock...it's obnoxious, but so is stealing your money.
Just remember to be polite, smile a lot, and be obstinate and politely annoying to the best of your ability.
Having a crying toddler helps exacerbate your dilemma...politely. You may also use the opportunity to change your baby's diaper on the ticketing counter...since everyone's being politely rude.
The airline will also attempt to issue a voucher, but these things typically expire within 12 months or less. If you paid with a credit card - that helps - you MAY be able to reverse the charge with your credit company, but that's also an exercise in patience.