Hello, I am James. Thank you for requesting my assistance. I will be helping you today. I do not get a salary here. But just like you, I want to get credit for doing my job. You are the only person who can help me get credit before leaving this site today. So please do not forget to do that by giving me a positive rating with 3 or more stars or any of the smiling faces.
Also, if you want me to be your personal immigration Expert, just write “FOR JAMES" at the beginning of any future, new questions, and I will be happy to help you as your personal Expert.
First, sorry for the delay. As far as your question, there is no reason why the US CAS should deny your H1B, simply based on a legal name change. Many people get their name legally changed, including through marriage. So this is not something uncommon.
As far as when is a good time, you can send the request now, while it is processing, or after you get the result. Make sure to send it with a certified copy of the order.
Kindly remember to give me a positive rating before leaving this site today. I do not get a salary here, so your positive rating is the only way I am rewarded for doing my job. If that does not happen, it is as if I never worked, which I am sure you agree would not be an ideal outcome. So please do not forget to give me 3 or more stars, or any smiling faces, in order to help me get credit for doing my job. Thank you!