My opinion - that IJN should've skipped on the whole 25mm AA gun idea - is not a secret.
A 20mm gun have had the advantage that it can be plopped basically anywhere on the ship. A 2-, 3- or 4-barreled installation is also easily doable, the 4-barreled installation probably requiring the same size & weight allowance as a 37-40mm automatic?
No disagreement on dropping the 25mm.
Japanese should had ditched 13mm or dropped it very early on.
A problem with some of these guns was the overly ambitious mountings. Maybe they OK when first designed/produced but aircraft design was advancing faster than gun/mount design and funding? Japanese test the 13mm guns in the early 30s, adopted it in Feb 1933 and got it into production in 1935.
US was using small numbers of .50 cal MGs and the British were using the .5in Vickers, mostly in a quad mount.
French and Japanese went a little overboard.
Four barrel unit was around 2600lbs but that is a bit deceptive as this thing needed a large crew. One pointer, one trainer, one sight setter and a number of loaders to deal with the 30 round magazines. Seven or more crewmen? And in a long engagement how far away were the ready ammunition lockers, where were the spare barrels and how long did it take to change them?
While the twin offered 1/2 the firepower it also was well under 1/2 the weight and used about 1/2 the crew.
The Japanese got this into production just about the same time they were asking the aircraft industry to start making monoplane attack aircraft. The large mount and complicated sight didn't actually add much to the effective range, which might have been OK against the B4Y biplane
not so good against the B5N that showed up in 1936/37.
Single barrel 20mm Oerlikons are a much better option than the 13mm and 25mm Hotchkiss guns.
Going past twin mounts for triples or quads may give diminishing returns. Even 20mm guns are going to start needing power elevation and traverse and reloading the center gun/s is going to be harder. And then there is the urge/desire to use better sights/fire control on the larger, heavier mounting.
40mm Bofors exists before 1930, and it is still my 1st pick.
The 37mm Breda, or even the properly navalized 37mm Flak 18 will still give them much better service than the 25mm.
The 37-40mm group of guns are a bit harder to judge. The Bofors was by far the best but there two different Bofors and a host of mountings.
Bofors came both air cooled and liquid cooled and came in simple had worked mountings, and powered mountings and in stabilized mountings.
Weight of mountings are all over the place

Type of mount and type of fire control could make a huge difference in effectiveness. In US service most of the quads and many of the twins had remote gun directors (although only a few meters away) to get the gun aimer off the mount and out of the smoke. US quads and twins were power operated and used liquid cooling.
Air cooled 40mm Bofors mounts with manual control are going to be a lot less effective although nowhere near as bad as the 25mm Hotchkiss.
British rarely used the 2pdr Pom Pom in single mount (or twins?) in it's modernized form. It lacked range but it had good destructive effect, a decent rate of fire, and was water cooled for long engagements.