The problem with Ratio changes is that OTL the US (and GB) are just massively richer so they don't have to accept anything less than superiority at 5v3 (and GB can just insist on septate totals for AUS/CAN/NZ/etc)
True, and the only way UK would likely accept something less is if they were completely trashed during WW1 and thus relied on having a solid alliance with Japan.
Well, it is not the first time, and will not be the last, that I'll bend real history for my convenience
Anyway, first entry of the never built Shi-Shi Kantai capital ships:
The first design for Koko no Kaigun postwar Capital Ship construction plan started to see light in early 1921, the Battleship's deign phase started under the guidlines dictated by two simple but precise requirements made by the Naval Staff: a main armament of eight 409mm guns, and a top speed superior to that of the Nagato and Tosa classes. Since for it first Shi-Shi Kantai unit Koko no Kaigun needed a 409mm-armed warships, but not as fast or as armed as the Kii and Amagi class beign designed in Japan, the new 2-units Battleship-class was modeled out of the Nagatos, but also implemented many improvements from the Tosa-class deisign.
The plan called for side belt to be kept at 305mm but inclined inwards by 15°. Anti-torpedo bulkheads were to be added below the waterline consisting of a 25mm plate backed by another 19mm one, increasing the full beam from 29,02m to 29,92m. Armour deck thickness would have been increased from 144 to 164mm split between two decks. Casemate, barbette, main gun and conning tower armor was to be kept the same. The hull would have been substantially lenghtened from 215,8 to 234,3 meters, enlarging both the foredeck and the quarterdeck for finer hull lines. Still, most of the lenghtening was planned for the enlarged boiler and engine rooms, for an increase in power output from 80.000 to 110.000 shp, which would have raised the top speed from 26,5 to 28,5 knots. Range was to be increased too, from 5.500 to 6.600nm at 16knots. Torpdo armament was to b kept, while the casemate battery would have been argumented with the addition of two extra mounts, for a total of 22 barrels. Newer 120mm/45 Type10 anti-aircraft guns would have replaced the four 76mm/40 type3 carried by the Nagatos. This Kai/fast Nagato design would have displaced over 37.000T, with a full load displacement close to the 40.000T.
The two planned ships were ordered at Toumachi Naval Arsenal and Yamatogawa Shipyards in October 1921, two months later than initially planned because of bureaucratic issues related of Koko's independence, but it was still planned to have them laid down at worst by January 1922. Instead, both ships construction was temporarily delayed when the Washington Naval Conference was called in November of the same year, then postponed indefinitely and eventually shelved in August 1922.