I've been developing a half-dragon gnome artificer named Pounce, and his whole thing is that he's a sort of magic toymaker in one of the cities along the Sword Coast. I don't know which specialty would go best with this concept. I think Battle Smith would go great with this, but I wasn't sure. Any thoughts?

Battle Smith would let you make a extra large toy robo buddy. Like Gato and Lucca from Chrono Trigger.

The other consideration would be to be artilirist and make little wind up toy cannons, "spring fountains" and clockwork snap dragons be your turret representations, never make anything bigger than the palm top size.

Battle Smith would free up some of your cantrips for utility cantrips like mendding which fits the concept

Alchemist I don't think fits at all. Its fun but feels rather awkward in play (for me).

So BS or Turrets are probably the most apt subclass. So kind of depends on how you want to fight.  Using a weapon and or heavy crossbow and or shield+hand xbow.  or Cantrip Blaster.

also.. half dragon gnome.? is that just backstory fluff-or is there actually a feature for mixing races?

Isn't a half-dragon gnome basically a kobold?

There's actually a template in the Monster Manual for creating half-dragons, so it's not unheard of in a mechanical sense—not to mention draconic sorcerer bloodlines—and the Volo's Guide kobolds are needlessly squishy as PC races go; at least compared to the Paizo ones. Pounce is kind of a spiritual successor for one of my old favorite characters that died (a mad genius kobold alchemist named Ymp from an old Pathfinder campaign), so you can imagine my delight when they released the Artificer class after I converted to 5e. It felt cheap to just straight up resurrect him, so I thought I'd do something new with his concept—and thus was Pounce born.

My first concept was for Pounce to make a huge toy robot to be a steel defender, but I gotta say, the wind-up toys sound pretty awesome; I can even imagine an action figure of an elf archer in place of a force ballista.