Background Check - Previous Employer

In a sticky situation and would appreciate help. Scheduling my final round at a tier 2 strategy consulting firm and am pretty confident I will get the offer, however, I start my new job at a Big4 next week (lateral position). It looks like my final round will overlap with my first week or two - haven't been able to speed up the recruiting process to secure an offer and renege before starting. I don't take reneging lightly but imo it's worth it.

I did NOT list my new job on my resume, and during interviews phrased my experiences at my most recent firm as current, although not explicitly so. I quit a month ago.

My options:

1. Only continue if I can accelerate the final round and get an offer before start date. HR is pushing back on accelerating even with "exploding offer." 
2. Start my new position, interview 1st or 2nd week of starting work, and don't mention this firm at all. Quit as soon as new offer is fully secured.

My concern is that my offer will be rescinded if my background check fails. Ideally I can take the first option, but the gap I didn't disclose to my interviewers will still be a potential problem. Do background checks find employment not list in resume? Will fibbing you're still working at a job you left a month ago result in the offer being rescinded?

Grateful for any and all thoughts!