Another option may be a credit union or small local bank that would consider doing it as a portfolio loan, but the rate and term may not be as good as conventional 30 year fixed.

I think they are just an extremely conservative bank and that is their interpretation of Fannie's guidelines.I have already gotten approved from some other banks (credit unions), so I'm not worried, but I am curious if anyone else has run into a bank telling them they need at least a 20% down payment on 3-4 units, regardless of it being FHA or not.

What matters more is whether the borrower in question has the (i) time, (ii) contacts, (iii) experience, and (iv) ability to effectively communicate & package their loan request to banks & credit unions in a way that is compelling to those lenders. 

I received wire transfer instructions to a credit union in Utah with a totally different account name than the company.

I do all of my commercial business with my credit union so I wonder if that has anything to do with it.

Check with credit union.

Talk to all of the local credit unions.

I’ve been working with my local credit union for the past few years, so I’ll head there. 

For anything less than that with a traditional bank or credit union, you'd likely be looking at a portfolio product (the bank/CU will hold your loan and not sell it on the secondary market). 

You might check with smaller outfits too like Co/Lab lending, Preferred Mortgage, Aceland Mortgage, and co-ops/credit unions.