Does the documentation she sent say she was leaving as of June 30? 

We also note the date the property was sold and the lender can easily verify recording of the documents. 

In any event, get them both to sign a simple document about how the money is to be distributed if an agreement can be reached.  

Also another reason people can need to pivot from utilizing conventional loans is their DTI gets capped or they are self employed and thus despite having ample liquidity to conduct the transaction they’d trying to do, can’t document income to the lender’s standards.  

Typically they will need to see 12 months of documented income from the property.

What documents do they require, what credit scores do they allow, how do they verify previous rental history, etc.?

If you can demonstrate that your year-to-date capital gains are in line with previous years and you have documentation to support this (such as brokerage statements and tax records), an underwriter might include it in the income average.It's crucial to work with a knowledgeable mortgage broker or loan officer who understands your unique financial situation and can navigate the complexities of underwriting capital gains income and 1031 exchanges.

Document everything with pictures and video.

HOWEVER, want to avoid/minimize setting up a new company and filing additional LLC and other related documents (want to avoid extra work additional docs would create). 

Do you have some kind of supporting documentation?