The FKM is neither particularly thin and flexible or particularly thick and stiff. It's a very nice knife for the price, but no longer the cheapest entree into high performance chef's. That honor currently goes to the Richmond Artifex.
The CarboNext is a nice knife and an excellent deal for the price, but it's real benefit goes to people who are already good sharpeners.
Wa knives have Japanese handles. Yo knives have western style handles. You've indicated a strong preference for "yo."
All in all the Tojiro DP is probably a better choice for you because it ticks all your boxes. The DP is priced in between the FKM and CarboNext. Like the Fujiwara FKM, the DP is entry-level to the high end of Japanese made western style knives but it does get you a lot of performance. It's construction is three layer laminate (called "san mai" in Japanese), which is something I don't like but I seriously doubt you'll notice the negatives. On the other hand, it does make the knife quite stiff, especially considering its weight and thinness. It's edge taking and holding properties are very good, but unlike the CarboNext the knife comes OOTB with an appropriate profile and sharp enough to use. The DP's VG-10 core is significantly less chip prone than most VG-10. Ergonomics, profile, handle comfort, bang for the buck, and F&F are all somewhere between good and very good.
I'd recommend a MAC Pro 9-1/2", but its a skosh out of your price range, and not quite as stiff as a DP (but stiffer than an FKM or CN). If you can afford it great, but the right length is more important than the extra benefits (handle, F&F, agility) the MAC brings compared to the DP. I also frequently recommend the Masamoto VG which has a better profile than the MAC but is a lot more flexible.
Most good knives aren't "balanced" in the sense you seem to think they are. Chef's knives/gyutos which are 210mm and shorter tend to be handle heavy; knives which are 300mm and longer tend to be blade heavy; while knives between 240 and 270mm tend to balance just forward or just behind the pinch point. There are a few exceptions; e.g., Viking/Gude are made to be handle heavy, and Globals (other than the "forged" chef's) are made to be balance neutral.
You're certainly entitled to a preference, but most skilled cutters don't care about "balance," considering it an "it is what it is" sort of characteristic. The sharper you keep your knives and the more adaptable your grip, the less it matters. The same is true about handle styles. As long as the handle isn't waaaaaaaaaaaaay too thin, too short, or too wide, a good grip makes wa as comfortable as yo, and vice versa. But again, it's a matter of taste not an absolute.
Sharpening and maintenance are incredibly important parts of knife skills. It's something you really have to lean into and will make ALL the difference in how you feel about your new knife after owning it for a few months.