Not so much archer as long range damage build.  What multiclass (if any) would you recommend?

For ranged Artificer builds - the obvious choice is either Artillerist (for spells) or Battle Master (for weapons). I wouldn't really recommend multiclassing because Artificers tend to get really great things every single level that would be a pain to miss out on. I'm sure there are probably some good options though - I just can't think of any.

A fighter Multiclass for a single level might be worth the investment just for the Archery Fighting Style and Second Wind... it also nabs you proficiency with Martial Weapons, since only the Battlesmith subclass eventually gets access to Martial Weapons on their own. If you actually start as a Fighter you can snag Heavy Armor proficiency, but I'm not sure just how useful that would ultimately be, since a Ranged attacker is most likely toing to focus on DEX, and high-DEX characters are generally better off using Medium Armor, even if they have the option to go Heavy.

That said, you can just snag the Fighting Initiate feat to get the Archery Fighting Style. Or possibly more useful would be the Sharpshooter feat, which is generally considered one of the most useful feats in the game.

The Battle Smith could work really well for ranged martial options. You can use your infusion for the Repeating Shot on a heavy crossbow removes the reliance on ammunition, lets you ignore the loading property, and makes your attacks magical. So you could pair this with the Sharpshooter feat and not even have to worry about taking Crossbow Expert. As a Battlesmith, you can also use your Int modifier for any attacks made with magic weapons (like your Infused Crossbow), which makes the build almost dependent on a single ability score. With your Steel Defender, you could also benefit potentially from a Rogue multiclass, as you can use it to help proc Sneak Attack as well. 

For heavy ranged spell attacks, Artillerist is probably the way to go paired with Spell Sniper 

I have played a ranged Battlesmith and can confirm what most people said. Repeating Shoton a Crossbow works great (a Longbow works great, too, but I'd call that a flavor fail for most Artificers). Having your defender as another melee blocker to keep enemys away works for the most part. As for multiclasses I'd only consider a dip in rogue or fighter. Both for the reasons mentioned before. Rogue for Sneak Attack, Fighter for the Fighting Style, Action Surge or even a bit of Battlemaster Action. That said you'd have to consider you're not only loosing out on the artificer stuff, but also your Steel Defender will be more squishy.

I've been thinking about this for a while. I want at least 2 levels of artificer for infusions and repeating shot. This is the easiest way to get a magic bow that never runs out of ammo. Then I want conjure barrage, which means 9 levels of artificer, 6 levels of lore bard, or the mark of creation and a 5 levels of a primary caster. So instead, I'll go fighter 7 and go arcane archer. extra attack, archery fighting style, bursting shot, entangling shot, and curving shot for the occasional miss. Yeah, you could grab 2 levels of fighter for action surge and archery fighting style and then go artificer-artillerist for more cover fire or defense while shooting.  

Quote from Obscureinfo >>

I've been thinking about this for a while. I want at least 2 levels of artificer for infusions and repeating shot. This is the easiest way to get a magic bow that never runs out of ammo. Then I want conjure barrage, which means 9 levels of artificer, 6 levels of lore bard, or the mark of creation and a 5 levels of a primary caster. So instead, I'll go fighter 7 and go arcane archer. extra attack, archery fighting style, bursting shot, entangling shot, and curving shot for the occasional miss. Yeah, you could grab 2 levels of fighter for action surge and archery fighting style and then go artificer-artillerist for more cover fire or defense while shooting.  

 I had a similar idea, though going the opposite direction with Artificer 4 (Battlesmith), Fighter 4 (Arcane Archer), then the rest Artificer.  That keeps you SAD with INT only needed. Flavoring Arcane Shots as Hawkeye-style gadget arrows would be cool.  Ideally, your DM would let you use crossbows with Arcane Archer, otherwise you’ll be using a longbow.    I’m suggesting Artificer 4 to take Sharpshooter early, and Fighter 4 to get a second ASI.

If you pick an elf, you can get Elven Accuracy, which is helpful for when you are going Sharpshooter.  Otherwise, you can go vhuman and take a pair of INT half feats for your first two feats.

I'd recommend Battle Master over Arcane Archer, mostly because Arcane Archer gives you two uses of your Arcane Shot per Short Rest, whereas Battlemaster gives you 4 Superiority Die per Short Rest. Most of the Battle Master Maneuvers are perfectly valid to use on ranged attacks... if you take it all the way to 7th level, you get an additional Superiority Die, giving you 5 uses of your abilities which pretty much all deal an additional d8 of damage in addition to a greater variety of options for secondary effects.

The Arcane Archer shots are on average more powerful than Battlemaster Maneuvers, but Battlemaster has a lot more up-sides compared Arcane Archer. The biggest problem with Arcane Archer is that the Shots aren't good enough to justify just how few of them you can use, which wouldn't be so bad except you only ever get two uses of Arcane Shot per short rest.

How viable is Alchemist Artificer with Arcane Archer or Battlemaster to make a gadget-enhanced archer?  (Not interested in the mechanical pet or artillery aspects of Artificer.)

I've looked through guides about the Alchemist.  Sadly, there's nothing in the subclass making them any better riggers of trick arrows than any other Artificer.  In fact, they are worse.