Reneging Advice - Before or after background check?

Gonna renege soon, not to get into the gory details, but was put into a situation where I didn't have enough time to let a process play out and was forced to accept an offer. Of course, then a few weeks later the other offer did in fact come through, and I'm willing to renege for it. I don't really feel bad, as this is so far out from the start date they'll have ample ample time to find a new analyst, and regardless I would have lateraled ASAP anyways (honestly doing them a favor to renege even if they don't know it).

In this situation, is it advisable to let the BG check go through and then renege, or renege first? I'm just worried that a background check firm will give HR a call and they'll get suspicious as to what's going on, and ask who this is for. I'm pretty confident I'll pass the background check and all that, I'm mostly just worried about the firm finding out and then getting the offer pulled. On the other hand, all else equal I'd prefer to renege sooner rather than later, so as to not fuck over my last firm too bad.