Welcome to Just Answer and thanks for your question. Since there is so little call for photos for British passports, you won't find any in the USA who specialize in that. I believe the reason most photographers have a problem is that they don't know what the British guidelines for photos are. You can download the link below that spells out the requirements and take it to any local photographer who does US passport photos so they know exactly what the rules are. Some camera shops even do them.
I gave the photographers to whom I spoke the specific requirements for a British passport photo and they advised that they only do U.S. so, in all fairness, your answer was not of much help.
Too bad. I'm sorry they wouldn't accommodate your request. You might try calling the British embassy and ask them for specific recommendations.