The Gadgeteer

[Arkhios]

I suggest Artillerist to at least level 5. That gets you Arcane Firearm; spell focus and damage booster to spells (including cantrips), but also a firearm usable as any other firearm, but still with whatever the wand/staff/rod you made it out of available too. In your case, a musket, which can be used with the Sharpshooter feat. Get yourself the Archery fighting style for +2 damage.

The Eldritch Cannon scales out fairly early (until 9th anyway), but as a bonus action, you always have a second attack.

Two level dip of Fighter for Action Surge, if you don't mind thinning spell progression. Going to 5 levels for Extra Attack may not be worth the spell progression loss.

An idea: Swords Bard's Blade Flourish works with ANY weapon, gets Extra Attack at 6th, helps with spell progression. Only the Artificer spells can be boosted with the Arcane Firearm, but slots are slots.

By my reading (and I read it several times over) Arcane Firearm doesn't specify a type of firearm, and it seems to me that it's not an actual weapon, capable of firing bullets, but rather what it actually is: a spellcasting focus. The "ammunition" used are the artificer spells cast through it. It works to an extent, sure, but it's not a weapon per se.

Archery Fighting Style gives +2 to Hit, but yeah. That'd work.

At first I thought that Eldritch Knight could be the answer because of "1/3 per level" progression for spellcasting, but I just realized that TCE Bladesinger 6 could be better, due to how their Extra Attack feature is written, especially with Artillerist: I could attack once per turn with the rapier, then substitute one attack with a cantrip and then use my bonus action to use Eldritch Cannon.

Since a rod/staff/wand is not an actual weapon, (IIRC) I could benefit from both Bladesong (wielding Rapier in one hand) and Arcane Firearm (a rod or wand) or Eldritch Cannon in the other.