Most LLM programs are designed for foreigners who already have law degrees in their country. The LLM simply introduces these attorneys to American law, and prepares them for the bar exam. The LLM programs for Americans normally do not accept Americans who don't have a JD. And, they are true masters degrees in the sense that they focus on a narrow area of law. They do not teach what the bar exam tests.
I don't know if it has been tried. But, an American could get a British LLB and then enroll in an LLM program for students with foreign law degrees. Then he/she would be able to sit for the bar exam.
I haven't sought out any of this because I simply don't have the money. I am going to stick with my paralegal certificate (I have 3 other degrees from accredited universities.), author one or two case digests, and start looking for a job after I pass the CLA exam. I plan to take it in May; if I get the $ 250.00 together in time. The University of Leiden is still up in the air. My proposal was on American law. I may be able to submit a second one on Dutch law. The challenge, I read online, is that Dutch court opinions are written in Dutch.