I have 2 factors that I know play into it, but there's always more beneath the surface.

Can anyone recommend a bank in Texas that will allow me to setup separate checking accounts for each Series of my LLC? 

This allows you to keep more money working for you in the real estate market, potentially helping you grow your investment portfolio faster.

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

On the other hand, keeping the house could allow us to maintain an appreciating asset in a desirable location.I’d love to hear from others who’ve faced similar decisions.

Most also don’t allow the property to be titled to an LLC/entity 

Secondly all DSCR should allow you to close in the LLC without a rate hit.If you are going to take out any cash through a refinance in less than 1-2 years be careful of how many years they try and hit you with the Pre-Payment penalty. 

The economics do not allow for a property renovation, properly permitted and all deferred maintenance issues addressed.  

They can separate themselves from traditional real estate investors by building a robust EIN Corporate Credit ("Ever-Increasing") investment infrastructure that allows for faster portfolio expansion and more sophisticated financial maneuvering.

They are structured via 506b so they are not allowed to market publicly (and only through referrals).