In the bible it says, "what goes around comes around." I see it here.

Boy a complicated situation. I'm not an expert in regards to the company's hiring procedure or immigration but one thing I can tell you that is resume changes over time. Minor changes are fine and even if you  provide false information in the application it can be corrected.

You can have, lets say a dual degree in Sociology and Software Engineering and if you feel that there is no need to mention your Sociology part of your degree while applying for software engineering position its fine.

If asked why didn't you mention you could say "you didn't find necessary to list Sociology because it irrelevant to the job I am applying."

I do not know how the immigration part works here so you might want to consult an attorney. It's better to do now before its too late.

Good luck.