How to Sort Bar Chart by Month Number from Live Query but Not Show On Chart

I re-created your problem and was able to get the X axis to sort correctly per your specifications.

Make sure the visual is sorted according to the Month and Year column as well.

See the attached .pbix  file, perhaps you can reverse engineer it to see where you went wrong.

///Mediocre Power BI advice, but it's free///

You are correct that there is no need to add it to the tool tip.

Did you download the .pbix file I created for you?  It has everything done, all you need to do is compare and find the difference between your file and mine.

Check everything to make sure you're doing it the same way.  Check the data types of the columns, make sure Month_And_Year  is sorted by year_month, and that year_month is formatted as a whole number, check the connection between the date table and the fact table in your semantic model, etc.