I strongly agree with you that lying about the IT background in order to get a job is really hard and it is hard to keep lying about it even after getting a job until you are well confident. But, we are in US and being in IT career is one of the best way of sustaining in US with legal status if you would not prefer to try the shortcuts like Army, Asylum and others.
Choosing BA or QA is very good idea as you do not have any IT background. I think you should go for it..Just take training from any consultancy and try to get a job.
I know it gonna be hard but better to get job and sustain your legal status than losing your hope.
Hope it helped you.