I know for a primary residence you need to live there for a minimum of a year unless there are some sort of special circumstances, is this true for a rental?

A lot of factors are creating some very unusual circumstances.

There are occasions where, in the future (but less than 12 months) if something changes your circumstances (such as getting a new job that you can't commute to) then you can probably get away with moving, because your "intent" at the time of the refinance was authentic. 

We were originally told they were doing cash for keys, which made sense considering the circumstances.

However there has to be a legitimate change of circumstance for change.

A smaller local bank or credit union would be more likely to fund under these circumstances.

If she would make a good tenant under normal circumstances (25-30% rent to income ratio amongst other things), then it could work..

Since these are on-market deals I'm talking about, and will likely involve my buyers agent, what are some points to make with her I can use to overcome what I'm sure will be seen as some kind of cognitave dissonance, since typically it would be unheard of to offer a house sight unseen (in normal circumstances, with a standard buyer).

Does this sound reasonable under the circumstances?  

Your side is a brand new shining shrine, while the rental remains the dump it was originally, because you neglected it for 10 years - which is unfair, by the way ;).