No, this would not work.
Assuming that this is referencing the Artificer UA
Additional barrels will not affect the game mechanics of the thunder cannon.
No where in the description of the thunder cannon is the number of barrels a factor nor is it specified.
The description of Thunder Cannon:
At 1st level, you forge a deadly firearm using a combination of arcane magic and your knowledge of engineering and metallurgy. This fire arm is called a Thunder Cannon. It is a ferocious weapon that fires leaden bullets that can punch through armor with ease. You are proficient with the Thunder Cannon. The firearm is a two-handed ranged weapon that deals 2d6 piercing damage. Its normal range is 150 feet, and its maximum range if 500 feet. Once fired, it must be reloaded as a bonus action. If you lose your Thunder Cannon, you can create a new one over the course of three days of work (eight hours each day) by expending 100 gp worth of metal and other raw materials.
The description does not specify the number of handles, triggers, sights, nor other accouterments that might impact the use a firearm as one might expect.
It states the mechanical characteristics of the weapon which accounts for the whole of the design whatever it may be.
The aesthetics of the weapons and equipment a character is using should be mutable so long as they don't
attempt to affect the mechanics in an end run around existing rules.