If you email them to ask, they are only going to come back and say no it won't impact your chances, even if it was to impact (which I don't think it will).
Remember a lot of the preparation needed for an AC is already done, You researched the firm when you applied, you have been building your skill set for many years etc. I personally feel like you could prep for an AC in half a day (many people may say this isn't enough) and that could easily be broken down into bite-size chunks over the next week or so. And most of that prep is reading things about the firm/market trends, familiarising yourself with key commercial topics the firm works with, and at best maybe a mock interview or practising some interview questions.
I think you have to do what is best for you though. If delaying your AC until next month is better for you mentally, then it's a no brainer in my opinion to move it to that date. You have to consider any risks that you are not in the right frame of mind for the AC and whatever else you have going on that day.