It would not be unbalanced, but it would be a house rule resulting in a buff to the player. Allowing it would be better than the War Caster feat's free somatic usage since the player also gains the benefit of the Arcane Firearm, but it's still insignificant in terms of balance because they can only have one action to either use the Firearm or swing the Sword.
You can have them make do within the rules which imposes some drawbacks. If the player has the Sword out and wants to use the Firearm, they can drop the Sword for free, pull out the Firearm as their object interaction, and then use it for an attack. This will result in them not having the Sword ready for Opportunity Attacks, and they'll have to swap back next turn. If they have the Firearm out they'll need to drop it to draw their Sword. In either case, having their alternate weapon on the floor could present a liability and limit their movement if they intend to pick it up. This is how I handle such cases.
As a halfway measure, you could offer the player a Quick-Sheathing Firearm that lets them draw or sheath the Firearm once per turn as a free action. This would have the benefit of not making them drop items on the floor but still limit their readiness for Opportunity Attacks based on whether they had the Sword or Firearm readied last.
I think it would be reasonably thematic and flavorful for a Gnome Artificer to create either the Quick-Sheathing Firearm or a Wand Sheath for themselves.