Go wtih what works - but intelligently.We offer a free month of rent, but to avoid desperate people, we credit 50% towards their 2nd month (to make sure they can afford 1st month rent + SD) and then credit the other 50% the 5th month.

Rate will be highly dependent on loan size, credit score, LTV, location, and property type, as well as a few other factors.

However, aside from that, I couldn’t find much information about them, and the only online presence seems to be the wife’s LinkedIn profile (though I’ve never met her in person and am just going off her profile picture on Zillow).After submitting the application through Zillow, we attached our income (two months of pay stubs for each of us from our current jobs), work references, credit and criminal background, rental history, and the contact info for our current landlord (who just so happens to be my fiancé’s father).

I've been self-employed for only 6 months, and loans are proving difficult to come by despite my great credit (not to brag, I'm moreso frustrated and seeking help).

Any local bank that does this kind of financing should be able to approve you as long as you have a good enough credit score and never declared BK.

If I have several options, I would prefer a HELOC with no initial draw requirements (i.e. to establish a literal line of credit that I have no specific short term plans to use), i.e. not something like the better.com HELOC. 

Once you start processing the credit/criminal, the money is spent, and you can't get it back.

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

Quote from @Jaycee Greene: Assuming you pay asking price, I'd estimate the down payment for something like that using a hard money loan (HML) would be around 40% of the purchase price and your monthly payment would be $2,600 with an interest rate in the low 7% range (probably need a 700+ credit score to get that).

The other way I fund deals with none of my own money is using a private lender that I have years of experience and relationship with and/or using HELOC and other business lines of credit.