Welcome to the site. Sorry that you had to find us all here and I am sorry to hear what your mum is going through as well, but I am glad that you have joined us all here as you're in the best place for support and help and I know that you will get loads of each from everyone here.
When my dad was diagnosed with his CC in 2008 it was deemed inoperable for the same reasons that your mums seems to have been in that it was located too close too major blood vessels and unfortunately this is quite common with this cancer. I too can not believe that your mum was told by a doctor that because the bile ducts are so small and a tumour blocks them quickly then that means that the symptoms show up at an early stage. If your mum really was told that by a doctor in a major cancer center then I would be horrified by that as this is most certainly not the case. For far too many patients, my dad included, symptoms do not start to show until the CC is quite advanced hence why it rules out surgery as an option for so many people.
It could certainly be the case that your mum is looking at things here through the rose tinted glasses or that she has not heard correctly what the medical team have told her and that is quite common as well. My dad did that a bit too if I was not there at his meetings with his GI team and I hope that you will be able to attend these meetings in the future with your mum. I agree with Lainy and Kris that getting more opinions on your mums CC is a good thing but if your mum is determined not to do this I understand the situation you are in. I am my mums carer as well and can so get where you are coming from with all of this also!
As Kris said, shrinkage through chemo leading to surgery at a future date is an option and some of our members have had that happen so please do not give up hope here. And keep coming back here too, we are here for you and know what you are going through. Keep doing your research and keep asking the questions as well. And please let us know how things go for your mum, you are not alone in this now.
My best to you and your mum,