What are the odds you can learn to deal with the noise? Any chance that learning to tolerate stuff like that would be an asset in the rest of your life? Just a thought.
I can see where it might be tempting to turn this into a 'she lied to me, I can't trust her' kind of issue. I'm not sure that's helpful. The sessions are at her house? The noise maker is her husband? In that case, it's his house too..... I have no idea what went in to her working at home situation. She may be in somewhat of a 'rock and a hard place' situation between meeting your needs and meeting his. Not your problem,obviously. It's just your choice how important the issue is to you and how you feeling about leaving now and either finding a different T or not. It seems like you need to weigh how useful is her help vs how horrible is the noise vs how important is it that you can control the situation.
I notice noises.They're more of a distraction that a trigger though, so it's different. But, every time the furnace kicks in, a door closes, the janitor starts the vacuum, an ambulance goes by with their siren on, what ever, I react to it. My T patiently waits for me to resume paying attention to what's going on in the office. Sometimes he comments on what the noise is. Sometimes we joke about it. In your situation, if I was seriously scared that the noise represented a threat, we'd talk about that and find a way to deal with it. (Because, in your case,it's not really a threat.) If I knew it wasn't a real threat and reacted as if it was anyway, we'd talk about THAT and find a way to deal with it. Because I'd find it useful to learn to deal with that stuff in other places too. But that's me,and my way of looking at it. We all have different issues and different approaches. You're really the only one who can weigh the pluses and minuses as you experience them.