Hey Kal,
If I remember right, you live in Tokyo currently but are moving to the US soon, right?
As you know, the Japanese have very stringent medical laws. It seems like the six month therapy requirement is pretty set and they aren't willing to negotiate. In fact, I've lost my hormones for the next few months simply because I won't be in Japan long enough to complete their therapy requirements. The doctors (in Osaka at least) haven't been willing to budge even though I have gone through therapy in the states and I was on my meds for four years. As for the legal age in which they will give HRT here, I believe it is set by the national government and the Ministry of Health and Wellness, so the doctors hands are tied.
If I remember, you said once that you were going to Minnesota? (Sorry if I'm mistaken, my coffee hasn't worked yet this morning). Informed Consent laws are different from state to state, so I'm not 100% about any states other than Alaska, as that is where I am from. But, when I looked into it, here's what I found: Dr. Meininger (the doctor you referenced) is actually quite beloved and is on several lists of providers, all with good reviews. There's another doctor in Minneapolis, Dr. Thorpe, though her clinic is a very small practice and doesn't often have openings. Both do informed consent, but Dr. Meininger was the only one I found that specifically has minors listed as a group he works with. His clinic is a bit on the small side, but it seems like he never turns away new patients, it might just be a bit of a wait. Another option is always the local planned parenthood. Some of these do offer Trans services, but not all.
All this being said, for many medications 16 is the age that they can be prescribed, even in the US. As I said, it all depends on the state. If you would like, you can let me know where you are going (PM is fine) and I will look at the local laws and clinics. My doctor back home before she retired wasn't on any lists for trans care. I just called clinics in my town and spoke with them until I found a doctor who would treat me. Finding doctors online is a good start, but if you know where you will be living, you will be able to call the clinics and doctors in your town and most likely you will find someone who isn't listed but is capable and able to treat you. I certainly had no complaints about my doctor even though she wasn't the "trans doctor" in town.