This is probably going to garner a few downvotes, but here goes:
If you have two conditional offers, then give them two conditional yeses.
Your goal is to get employed, so here's the decision matrix:
@Mircea offers the alternative option of telling them "yes, but" and proactively disclosing your "impropriety". This is basically the same as the "both checks fail" scenario, only you're hoping that the company will be lenient because you're coming clean beforehand. This seems pretty starry-eyed to me: companies don't like risks, they've got their criminal/credit risk screening process in place to reduce those risks, and it would take a brave hiring manager to override that process, because he will be blamed if something goes wrong. In most larger companies, the manager would not even have any discretion here.
Also, as far as ethics is concerned, this is not something like fabricating a diploma out of thin air, the worst you're committing here is a sin of omission, and you're dealing with a job offer, not a marriage proposal. If you're never asked whether you've committed an "impropriety" and the check doesn't pull it up, yet it still somehow comes up later, simply stating that you didn't think it would be an issue is perfectly defensible. (Although this may not stop the company from firing you.)