@fixitwithahammer
I'm very sorry that you were subjected to such poor service, especially with a baby in tow.
Whenever you deal with front office staff and they become negligent, abusive, or just are NOT doing anything related to customer service, escalate things immediately to a supervisor, and keep going up the chain until you get things done.
You can also threaten to call the police (and don't be afraid to do it), for theft, fraud, etc. In this case, you'll need to have the airport police come out, then request the formal police come out, so you can file a fraud and theft report. Although nothing will come of it - filing a police report against staff is still a serious issue and will cause the front office staff serious problems with their management all the way to the top.
NOW it will be their problem. Remember to save up this trick if the supervisor also is reticent to help resolve the problem.
Make sure you document EVERYTHING legally, including names of the front desk staff, the supervisor, the time, etc. You will need this information for the police report.
This will give you something better to do with your time, than trying to negotiate with negligent, poorly trained staff and it will give you the satisfaction of knowing that you are a very courteous person - you can also shovel hot potatoes around.
In all things - remember to be polite, patient, firm, and matter of fact about regretting the need to file a police report.
The police report is necessary for filing a civil fraud claim against the airline, when you return home. It will include dates, times, names and is an official document (don't forget to get the police report number).
Most front desk staff think customers are powerless, but we're not - you just have to know how to empower yourself, and be prepared to use that hammer. Since they're obviously lying to you - you can also make up stories scary stories about what happens to front office staff when they fraudulently misrepresent transactions that cost money (fired, jail, and the follow-on civil suits for compensation, which they'll now be solely responsible for repaying, as they've been fired, which will also add weight to your civil and criminal case against the airline.
This is just for future reference - as you NEVER know who CE has a code share agreement with...so know your rights and how to be an utter stench.
If you're elderly - you can also do the "grab your heart" thing - for maximum melodrama - maybe even get the media to show up - they love a scandal.
Just make sure you're absolutely in the legal right, read your ticketing information legalese carefully.
Also - make sure you have the corporate office's number, as they need to know that a huge stink is coming their way, with that police report.