Though this may have been put forward before, and it may not fit entirely into the game unless it's added as another type of 'antique' weapon you can get from Jelly Gems... Shotguns. And I have actually fleshed out a possible trio of trees.
The gun itself would fire many rounds (a la antigua travelspeed but with less damage) that deal either normal or piercing damage, that single part I haven't thought on.
The regular shotgun, a 3-star weapon, may have a name such as Regulator, Campabella or Thundermaker. The closer you are to your opponent, the more chance you have of hitting your target many times for a decent amount of damage, with only two rounds per 'clip' and knockdown chance on the second shot.
The Charge attack would simply fire both shots at once but less precise, making it decent but not ideal for crowd control, each opponent struck may be knocked back.
The three improvement paths would be as follows: a fiery path that combines elemental damage with normal damage that has the same capabilities as above, but with a chance to set your targets on fire. Its five-star upgrade would simply be an enhancement on that, as well as a gout of flame at point-blank range that has a high chance to set your targets on fire... but it's a one-square range.
A piercing-only path that is quite like the Callahan, firing large slugs, but with a much larger (5-shot) clip. Less damage per shot than Callahan, but you can move while firing, and the five shots set it aside from that particular gun very well. Only the charged shot would have knockdown though, and its range would be slightly shorter than the Antigua line.
The third tree would be normal damage only, but just an improvement on the basic gun itself: more shots (four or five per clip), more damage, less spread. A typical hunting shotgun with a narrow choke. Buckshot, if you will, but all in all each pellet deals considerably less damage per shot (say 30% of AP damage disregarding vulns/resists, but with 5-8 (random with each shot) inaccurate pellets being fired.)
Fire rate: average unless you take the piercing damage tree, then medium-high. Damage: below average per pellet, high total. Medium/high on piercing tree.
Reload rate: Sloooooow. Slowest of any gun out there.
Range: short-medium, slugs medium-long but inaccurate.
Elemental effects: only on the fire tree, and then looooow chance per shot. High only on charge burst.
Upsides of this weapon tree: rewards gunslingers and other shotgun enthusiasts for being closer to their opponent, trading safety for possible damage. Knockdown, crowd control.
Downsides: rewards gunslingers for jumping right within their opponents' faces, making them juicy targets, especially with the weapon's slow reload speed. Also cannot fire while moving unless using the Piercing tree.
This may or may not have been suggested before: if it has, well, sorry. If it hasn't, hey.
Firstly, suggestions forum.
Secondly, you just described the autogun line with a new charge attack (autogun fires a large number of bullets over a spread which is more effective at close range - it's basically a slow shotgun. The needle lines even knockdown, and the pepperbox has a unique crowd control ability). Even down to the variants, mostly - normal+ fire = pepperbox. Piercing = needle gun.
Also of note is you suggested major gameplay differences betwween the variants - it's counterintuitive for players to essentially be forced to learn to play with a new weapon, when most other weapons mostly play the same throughout the line (biggest one is your piercing variant, which suddenly turns from a heavy gun into a light gun)
The biggest issue with making a shotgun is that it's going to feel like a faster autogun in terms of how it'll be used. Honestly, I was expecting the autogun to be more of an SMG when I first saw it in the weapon shop, and not a delayed shotgun. Would've been pretty cool if it was - a gun that fires for as long as you hold the attack key or until reload... ah well. The problem here isn't with shotguns - it's that the game's only automatic gun plays too much like one.