In order to get an ITA a month before your actual job completion date you would have to adjust your employment time elsewhere and in this case you would have to adjust your job start date. So, let’s say that in your EE application you have mentioned your job start date as August 1st but in your employment letter it’s mentioned as 1st September.
Now an officer reviewing your case (who is having a bad day) can construe this as misrepresentation of facts while some other officer (having a very good day) can let it slide after reading your LOE. So, I’d say it depends totally on the officer reviewing your case.
Personally I don’t think it’s a risk worth taking, consider the fact that you will need to wait for about 6 months after your submit your application, even if the officer ultimately approves your application you would keep worrying every single day of that waiting period, which I feel is far worse that waiting for another month or so.