I lived in Japan for 2 years (just got back) and went through exactly the same thing, especially wondering how the heck JApanese people are so skinny with all the white rice they eat (they actually will often claim that's the reason they ARE thin) and the glaring lack of vegetables (I would eat raw veggies at work and everyone told me not to because it would mess up my stomach). The biggest challenges came when I was traveling because I wouldn't have a kitchen and as you know, eating out in Japan is never that healthy.
I actually gained a bunch of weight in Japan so I'm not very helpful but when I did try to do the SBD there (and fell off it every time I traveled), here are some tips.
-be careful about udon noodles, 100% buckwheat ones are actually really hard to find
-MISO SOUP! This was my staple breakfast food and it's totally SBD-friendly. They sell low-sodium miso paste in most grocery stores.
-For eating out, one of the closest things I could find to SBD P1-friendly was yakiniku. You grill it yourself so you know what's in it
-Learn the kanji for wheat and sugar and check ingredients
-Learn to give up on looking for nutrition info. It's still not a big thing in Japan
-ORDER STUFF ONLINE! fbcusa.com was my savior for things like beef and veggie broths and low-sodium options. Also for tomato paste and sauce.
-Be ok with splurging on imported food AND with a slower SBD progress. There is no such thing as whole wheat bread in Japan, nor low-fat cheese.
-When you do go out to eat, try family restaurants like Coco's. They have SALADS *and* a great cheap dish that's just chicken covered in pesto that's pretty yum
-Give up conbinis! The only SBD-friendly things they really offer there are nuts, hardboiled eggs, water, diet soda, and at some of them, those Hokkaido cheesestrings. I was a huge conbini junkie so giving them up was one of my biggest challenges.
Haha sorry these tips are a little bleak. I had a really rough time with SBD in Japan. If you are set on losing weight honestly I think it might be easier to switch diets than to go through all the stress and hassle of trying to stay on SBD. Good luck!