Ive got a player wanting to make a lvl10 wild west/John Wick style gunslinger type character (with making a Mercer Gunslinger) using hand crossbows. With the first 3 levels as Artificer and the remaining 7 as battle master, with crossbow master as the vuman feat and +1 crossbow and repeating shot as the infusions, which artificer subclass would you use to begin with, Artillerist or Battle Smith?

Battlesmith is better for weapons users because of its ability to use INT for weapons.  Do note that the repeating shot infusion can’t go on a crossbow +1 or Enhanced Weapon infused crossbow, although it is a +1 on its own.

They can flavor their spells as trick arrows if they want to lean into it.

Yeah both infusions are on seperate weapons, so both could be used with the INT mod ability.

I'd say ditch the gunslinger multi-class and make him a wandslinger using Firebolt as his go-to attack and go straight Artillerist.  Artillerist is going to want to have their bonus action freed up for their Eldritch Cannon so Crossbow Expert with hand crossbows are not ideal.  So in theory you can have them with a wand in one hand casting blasting spells and their Eldritch Cannon in the other using Force Ballista.

I would just go straight Battle Master. At 11th level, a Battlemaster with Crossbow Expert & a Hand Crossbow, it’s like they have a Mac10. With Crossbow Expert they don’t need Repeating Shot at all. And go as a Thri-kreen from the UA and they can hold a shield and a sword and a hand crossbow and reload it all at once. Can you imagine human-sized Mantis Person with an SMG crossbow and super powers? That’s what it’ll be like. 😉

100% Battlesmith over Artillerist. Although Artillerist seems thematically more appropriate for a gunslinger-focused character, an artillerist is actually designed more for spellcasting than attacking. Artillerists get Arcane Firearm, which increases damage of Artificer spells, but which has no synergy with a character who's going to focus on ranged weapon attacks, whereas Battlesmith gives Extra Attack, the ability to attack using INT, and their Arcane Jolt damage boost can be tacked onto ranged weapon attacks.

Artillerist would really only be potentially more useful (or at least more thematic) than Battlesmith if you're planning to take just three levels of Artificer and focus mostly on Battlesmith, with Artillerist just being taken to get access to an Eldritch Cannon, but ultimately the Eldritch Cannon alone isn't particularly useful, especially since you've already got Crossbow Expert to make bonus action attacks round after round at no additional cost of resources. It gives you a little more variety, but at that point you'd probably get more benefits from just going full Battlemaster.