Where did you get this information? I'd like to research the third class further.
One to try is George Behe's (a top historian of the Titanic disaster) book "Voices from the Carpathia". This book includes dozens of testimonies from her passengers and crew. You find out what they went through and the confusion and hurried reorganisation on board the Carpathia as they prepared to receive survivors and what happened when they finally did.
There is also Captain Arthur Rostron's evidence to the enquires in the USA and UK which you can read for free online.
The whole crew of RMS Carpathia did everything they could in difficult circumstances. They deserve a lot of credit for their hard work on April 15th 1912.
One thing I can pretty much positively assure you of is that no woman with a baby in her arms would have been turfed out of her cabin and forced to huddle up against the funnel for warmth.
The British Parliament had passed strict laws regarding British ships carrying immigrants in the years preceding the Titanic disaster relating to how they should be accommodated and looked after. RMS Carpathia was a British ship and had to abide by these laws.
Premier North Atlantic immigrant carriers such as Cunard and White Star obeyed these laws to the letter. They didn't want the terrible publicity, fights in court and large fines for breaking the law.
Your relative was likely woken up in the middle of the night and bluntly, without explanation, quickly moved from cabin X to cabin Y. It would have been a confusing and upsetting experience but when the morning broke and she saw the Titanic's lifeboats coming alongside she would no doubt have understood the reason why.