Definitely order her FBI check. If it is clean, I would suggest you apply for her PR now. Could you get a loan from family to cover the cost?
You have a lot of proof that the relationship is genuine. The problems are the potential criminal record on the FBI check. The other red flag is that she has entered the USA illegally - CIC often sees that as evidence that the applicant is desperate to leave and so will suspect a marriage of convenience. However, I think you will have enough evidence of a genuine relationship to overcome that.