Seeking Advice: Difficult Exit Scenario (Consulting -> IB Corp. Strategy)

Recently received and accepted a verbal offer for IB Corp. Strategy opportunity. Due to the hiring firm's system upgrade, there's been some delays in starting background check which I assume will take ~2 wks. Given my tentative start date is 2nd week of April, I expect background check to clear a week before start date which puts pressure on giving my 2-week notice much earlier. 

To top that off, I was just staffed on a high priority project with CEO's direct visibility which will require biz travel in a few weeks. I have a respected relationship with my direct superiors but I know there's no way of avoiding burning bridges when I give notice. Having said that, I am looking for advise on how to exit as smoothly as possible given the circumstances. TIA