I also recommend, if you are unable to find another tenant that meets your NTN requirements, simply let this tenant know there are a few factors that show higher risk on renting to him/her, and then simply ask "Are you able to put up one extra month's rent in advance and maintain rent on time?"
This would allow you the additional peace of mind, and it would allow your tenant to get in. (ultimately this is simply increasing the deposit and that;s how it should be written up on Lease, however, if you present it in this way, it may be absorbed easier)
If you and your land lord are still unsure on the possible tenants credit history. I have been known to asked them in the past to provide a "current" copy of their credit that the can get for FREE on annualcreditreport.com. This allows my landlord the opportunity to see how current or past the issues are. Hope this adds some assistance to your question.
I have been discussing the issue with LL when listing a property for lease. It is becoming more and more common to find major *glitches* in financial histories.
If previous rental history is good & there is verifiable income, it is possible that the prospective tenants could put more deposit down to boost your comfort level. I have even had LL agree to a double deposit w/ 1/2 of dep. to be credited to last month's rent so long as all rents have been paid in full and timely.
Hope that helps.