Scared about background check

Hello, I just graduated last month and started working at a very big company two weeks ago. I hate my job, and I am interviewing at other places (small independent firms and banks for ER). On the background check, can I omit that I am currently working? I want to quit the job as soon as I accept an offer. I also stupidly realized that I made a dumb mistake and put BS instead of BA on one of my majors on my resume. Should I email them now with an updated resume? I have an upcoming second round with them. Should I wait until they hire me? My transcript kind of sucks with some C's scattered around and although I have a decent overall gpa (3.5 - 3.6) I would like for them to not ask for it until I get an offer. I went back and reviewed my resume and noticed that the big company that I'm working at did not notice the discrepancy between the BS and a BA but this smaller company might do some of the checks themselves, as they are very detail-oriented about the resume. I know these are a lot of mistakes and I appreciate any insights/comments you have!