I had worked on something similar a while back, but for crossbows at first. I wanted to have crossbows be simple and have firearms be the martial equivalent. Since combat is seemingly balanced around 3 rounds, I was trying to balance it so that if someone did Fire, Reload, Fire as their 3 rounds, it would be equivalent (which would mean the dpr target would be around 1.5). I also wanted to ensure it was 1 shot per attack, regardless of extra attack, and to have cantrip damage scaling.
After polling my players, the majority said they'd only use it for a 1st round shot and then would wait to reload, even if it was an action. That makes it 1.5x, 1x, 1x, would would just be a buff. The majority said giving up a whole action to reload was a non-starter.
I do also agree with those here that are saying they don't think you should have these be a Dex save. I totally know what you're going for, but plate armor was often "proofed" against bullets, showing off the dent where a bullet didn't penetrate ("bullet proofed").
If you want a mechanic that could be fun and unpredictable, you could do something like "roll 2 attack rolls, if both miss you do 0 damage, if one hits you do half damage, if both hit you do full damage, add 1 damage die for each attack that crits."