Several things come to mind.
The first thing that comes to mind is the question: “What do musical theater, veterinary medicine, pregnancy, and the Naval Academy have in common?”. The answer is that you really need to be fully committed before you take on any of the above. You need to be sure that you want to do it. I do not think that committing 9 years to the navy is something that you do unless you are very confident that you want to do it.
The second thing that comes to mind is that I am confident that our armed forces are happy to enroll people who have graduated from UCLA, assuming that they have an appropriate major. Mathematics, computer science, or almost any type of engineering come to mind as some examples of majors that would be potentially interesting to multiple branches of the US military.
The third thing that comes to mind is that $45,000 per year by US standards is actually quite reasonable for an education of the outstanding quality that you would get from UCLA.
And I do think that this is an issue to discuss with your parents, or just go with UCLA.