I have programs that require only 10% down for the purchase, and will finance 100% of the rehab.

You can probably make up for the perceived loss through the reinvestment in your US business.

As mentioned by @Chris Mason, Bridge Loans: These short-term loans can provide immediate financing until more traditional financing becomes available or until the property stabilizes and becomes more attractive to lenders.Interest rates and stipulations may be a pain.Out of the box here and requires a few more research, SBA Loans: Small Business Administration loans can be an option, especially the SBA 7(a) loan program, which can be used for hotel acquisitions.I heard someone  mention - Mezzanine Financing: it involves a combination of debt and equity financing, which might appeal to some investors looking for higher returns.

I recommend you research the local section 8 program. 

When you pay cash for a $400,000 home and rent it out, your rate of return will likely be less than 6%.You could invest in stocks and earn 7% or better.You could put the money in a high-interest savings account and earn up to 5%.I don't understand why anyone would pay cash for real estate, erasing many benefits that increase the rate of return, while exposing themselves to the risk of loss (turnover costs, bad tenants, capex, etc.). 

Hopefully that upfront diligence plus a healthy contingency budget would suffice to keep me out of that scenario, otherwise I would foresee a guaranteed loss, or underperformance at the very least.

You should fire them right away and talk to an attorney to prevent the loss, but I wouldn't expect much and it could be a waste since a company that does this will not have the money to do anything.When you have just one unit and you are looking for PM companies, your options are the bottom of the barrel because it's not a good use of the big one's time.

What other programs do other people use to get addresses to dispute these? 

The 25% has to be your funds.Now, there are non conventional options that would be a bit more flexible but even most of those programs will require at least 20% down with no second mortgages filed on the property. 

There are a few programs that can do second position financing but typically underwritten based on your personal income.