Big Bank vs Small Banks/Credit Unions

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Would it be easier to get a low down payment loan (like an FHA) from a big bank like Chase, who I've been with for about 4 years, or a local bank that I have no relationship with?

I've heard in the forums and on the BP podcasts that it's usually better to go with a small, locally owned bank or credit union. I want to get a FHA loan to house hack my first property, but I don't have a relationship with any of the smaller banks yet. But I have been banking with Chase for a while and see they offer an FHA loan. I'm not sure which route would be the most efficient. How much does time spent with the banks play a part in them working with you on a loan?