We can look at the one historical cartridge and several modern cartridges that will fill the bill.
We also need to look at the actual requirements. Getting a small bullet to 200 meters is easy. Having it get there and do what is wanted is another.
Soviet 7.62x25mm, French 7.65 Long, Japanese 8mm Nambu, British .380 Revolver, German (and others) 9mm Parabellum, U.S. .45 ACP and British .455 Webley.
As can be seen, only the French 7.65 is actually smaller in case size and will allow for smaller magazines or larger capacity for the same size magazine.
French used a 77 grain bullet at about the same speed as a 9mm Parabellum fired a 124 grain bullet. The 7.62 Soviet(7/63 Mauser) uses a 85-90 grain bullet about 60m/s faster.
Some soviet Sub machine gun rounds may have been loaded hotter.
Going smaller than 7.65 means even lighter bullets and short stumpy bullets don't do well at long range, Long range here being 150-200 meters.
S&W and Federal cartridge have teamed up for the .30 Super Carry
of much better performance. But it uses a bullet closer in weight to the standard 9mm bullets and it operates at military rifle pressures. You need good tolerances and good materials in the guns.
You can use a bottle neck case with 5.56-6.5 bullets but you are going to need high pressures and even with pointy bullets the 150-200m performance may be lacking.
.45 Auto, .40 cal, 9x19mm and the FN 5.7x28 used in the FN MP90 and several pistols. The pistols have long, awkward grips.
The 5.7 bullets are around 30 grains but the commercial ammo varies as some of the Military stuff is considered armor piercing against body armor and is illegal in many countries or US states.
Hope this helps