When you use a sword you also put prof points in your fighting style (single/double/two-handed). Are any of those appropriate for a bow user?

Thanks

No, they give no benefits to missile weapons.

I often take Single Weapon Style or Two-Handed Weapon Style, though, for a bonus if I switch to a melee weapon as backup.

While I think that the more rarely used 2 handed weapons are great weapons for an Archer (why give the most popular melee weapons to an Archer who is trying to NOT use melee weapons), I think that it's arguable that Single Weapon Style is a good style for an Archer, since it gives an AC bonus with only 1*.

Does one receive that AC bonus for Single Weapon Style when the bow is in use, or is it only when the melee weapon is being used?

Only when the melee weapon is being used.
-Brainless One

Bows should get a benefit for 2-handed Weapon prof IMO

But, if I understand correctly, there is currently no weapon style which gives bonuses to an archer when a missile weapon is equipped? Sorry, I may not have been clear before.

If there isn't, I think there should be. Or maybe it could be an HLA, 'Steady Hand' or 'Eagle Eye' or something. Its not that I think the lack of such a style is a detriment to the archer, they're still great. But once I grand master the bow I want to use, which you can do right off the bat if you want to, I keep getting points and eventually I just have to put them into weapons I'll NEVER use. It seems a waste.

Maybe a total of five different fighting styles then? Sword & Shield-style, Two-Handed Weapon-style, Two-Weapons style, Single Weapon-style, and Ranged Weapon-style?

I agree.

If you want to go for Grandmastery in some 2 handed weapon, you can actually spend 7 points. 5 on the specific weapon, and 2 on 2H style.

But you don't have that option for ranged weapons. The idea of a ranged HLA is interesting, although I don't know what it could be. (Do the existing HLAs whirlwind and greater whirlwind already work with ranged attacks? I don't know. Probably does, but I've never verified it.)

The existing Critical Strike HLA would be a good ranged HLA, except that it has 2 prerequisites. I suppose that someone could create a new ranged only version of it that didn't have such requirements, although it might be a decent idea to require grandmastery in a ranged weapon, like bow, crossbow, or sling.

Another possibility might be to add a new innate kit ability to the Archer kit, available at some level above 20 or so.

Or extend the existing Called Shot ability (it tops out at level 16).

There are a number of things that could be done. However, the Archer is already one incredibly powerful class and additional power seems rather superfluous except for a bit more flavor or ambiance to the kit.

Correct.