The 750 hours requirement you are referring to is in relation to whether a taxpayer qualified for real estate professional status or not.

Would they have to qualify with a lender as well? 

The way it works is you first find a tenant who wants to buy a home but doesn't currently qualify for a mortgage.

If it were me, I would work with them and try to find a "well qualified" replacement tenant. 

You want to work with an investment team that already has the skills and a proven track record of finding, qualifying, renovating, and managing properties.

We spoke with our mortgage lender who said we could look into qualifying for both homes, and do a cash out refi on our current residence... but we would have to wait 6 months to buy another property.

VA Entitlement Restoration: Veterans can restore previously used VA entitlement by:-Selling the original property, repaying their current VA loan in full, and disposing of the home-Allowing a qualified Veteran to assume their current loan and substitute their entitlement for theirs-Refinancing their existing VA loan into a non-VA product and invoking what's known as the "one-time restoration of entitlement."

Other than buying a property with cash or going with a hard money lender, are there any ways we could qualify for financing?

We always look at gross income to qualify.

He doesn’t even qualify to schedule a showing in my criteria.