It's not likely that a bank would refuse to sell you a property because of the type of loan you are using. Instead what a lot of banks will do is require you to go through the loan pre-approval process with either them or a lender of their choice. That doesn't mean you are required to close with them but they do want to make sure you are a qualified buyer.
I can't say I blame them. Many loan officers give out Pre-Approval letters without doing little more than just speaking with a potential buyer on the phone. I got a new client a few days ago that was trying to go under contract but at the last minute his lender said he didn't qualify after all and needed a co-signer AND 20% down. This happened through one of the big banks that is now owned by Wells Fargo and I will leave it at that.
There are plenty of good and bad loan officers out there and it doesn't matter if they work for a broker, banker or large depository lender.
Most brokers, bankers and direct lenders can offer the programs you are looking for. What you need to look into is how their Pre-Approval process works. They have to take a full application from you, review credit, debt, income, assets, submit your file through an Automated Underwriting System and if need be clear it with an underwriter if there are serious issues.
There is a lot more to it than that but if you would like more details feel free to contact me. Thank you.