How to cover nail holes in composite trim in a wet location

Less wiping.

More time before sanding.

Unless you go to something extreme like epoxy putty, it all shrinks. If you leave the filler proud of (sticking above) the hole, let it cure fully, and then sand it down, it should work. In extreme cases you may need to refill and let that cure, but that's adding more time to the program which is probably not good for you. If you scrape it flat to the hole, it will shrink below the hole. Without adding any overall time to your program, I'd save sanding for the next day just before painting, at least. Excess putty near the hole will come off with a sanding block (and if you don't use a block, that can also promote divots) - since you specified that you are painting over, filler beyond the hole should make no appearance difference if it's allowed to cure and sanded flat.

If the building is not fully heated when applying the filler, or is heated but is dumping a huge amount of water from fresh drywall work (or unvented construction heaters), that may affect the curing time of the filler product negatively.

I have personally had better luck with "modern" "lightweight" filler compounds than with "wood putty containing actual wood fibers" (which, inherently, shrink quite a bit, because they are wet wood fibers...) - the fact that the nailholes are glaring white .vs. the wood surface does not matter if you are painting over them. Given that you are (I gather) in a production/professional capacity, probably best to run some experiments with different fillers and schedules and see what works best for you.

The "mysteriously gigantic" nailers are probably 15 gauge trim nailers similar to this one:

Sounds like the one that you have is a finish nailer, and is likely a smaller gauge and uses nails with smaller heads. Thus less driving power required and a smaller gun. A trim nailer will generally give you better holding power, but also higher possibility of splitting wood, especially thinner stock.

Assuming the nailer you have is working for what you're using it for and can use long enough nails, I don't see any reason to stop using it - I've trimmed with a finish nailer before without issues.

As far as your metal tip goes, I would doubt you'll be able to find one from the manufacturer for a cheap Harbor Freight gun. One thing I've done in the past when I've lost mine or broken it is to lightly dip the tip in Plasti-Dip (the rubberized tool handle coating stuff). Just make sure that it doesn't interfere with the action of the gun at all 1/16 - 1/8 inch is plenty.