I did a similar thread based on this thought because I suspect there is a scientifically "ideal" battleship design much as during the Age of Sail the "seventy-four" third-rate was the standard warship because of the balance of speed, maneuverability, cost, and firepower (and I suspect in the Cold War it was the 10K ton CG). The reason we never really hit on such a design is because the dreadnought age was barely 40 years and the final generation of battleships, the nuclear-powered battleship (which would be akin to mid-19th century ships of the line which used steam power), never emerged. I think it would be extremely challenging to both find such a standard and have it produced on a large scale in numerous nations before the battleship is obsolete.
At one point I was thinking a Yamato/Montana-class ship might be your standard battleship, but the cost and just as importantly the size (needs to fit your docks, need to fit through the Panama Canal, etc.) would serve as a constraint. Because the United States is going to be a major player in this and needs to worry about Panama Canal, this means the standard would be close to a Panamax ship which would be a slightly longer Iowa-class with more draft. The standard gun will be in the 16-18" range. I favor the lower end because IIRC 18" guns have some inherent issues due to their size and the technology of the era, are more expensive, and have slower rate of fire.
You will see a few larger ships to counter them, but they're so prohibitively expensive nations will build very few of them, and even nations which could build them may prefer not to in favor of more smaller ships. I think given technical issues, you won't see anything much larger than a 20" gun ever built. Realistically even those might be sidelined in favor of packing a ship with more 18" guns. Among these ships, you might have an initial generation of true super-battleships and then attempts at building "super-super battleships" which by that point might be nuclear-powered and likely armed with missiles and not big guns. The "super-battleship" would be your Yamato/Montana-class at over 70K tons and the counter to that might be around 90K-100K tons (like the Nazi H-42/H-43 plan).
Carriers are difficult to invest in because they aren't proven weapons in the interwar period and there's a considerable barrier to entry lest you build something like France's Béarn or the Nazi Graf Zeppelin. It needs the planes, the cooperation with other branches of the military (potentially), the doctrine to properly use carriers, and properly trained pilots. Smaller nations would be skeptical about buying a carrier.
This happened OTL, Spain built 3 very small dreadnoughts because the government had little money but demanded ships. It's the exact same as how in the 18th century Portugal, Venice, and other small nations built third-rate ships of the line. Battlecruisers might be popular too among these nations (see the Dutch battlecruiser proposal) since their fleets won't be expected to clash with the major powers and battlecruisers do a good job at showing the flag and hunting enemy surface raiders and winning smaller engagements.
As for a rating system for ships armed with guns, here is my concept:
*First-rate - "Super-super battleship", 18-20" guns, over 80K tons
*Second-rate - "Super battleship", 16-18" guns, 65-80K tons
*Third-rate - Battleship, 15-18" guns, 40-60K tons
*Fourth-rate - Small/obsolete battleship, battlecruiser, "cruiser killer", 11"-15" guns - 20-40K tons (akin to how fourth-rates were no longer considered ships of the line by the end of the 18th century)
*Fifth-rate - Heavy cruiser, armored cruiser, 8"-11" guns - 10K-20K tons
*Sixth-rate - Light cruiser, 5"-8" guns - 5K-10K tons
I think we could've seen the basis of this in an ATL without the naval treaties and in some ASB world stuck at 1910 - 1950 technology for centuries this would inevitably emerge.