To finely spray a thin layer of warm liquid Coconut Oil?

(this is more a comment, as I haven't actually done what you're asking, but based on what I know ...)

I've used two types -- there are ones that you pressurize and then it works like an aerosol, and there are ones that are just your standard pump sprayers like you'd have on bottles of cleaning solutions where each trigger pull pumps a bit of liquid..

I prefer the pressurized ones, although over time they can get gummed up and may require a thorough warm soak to clean so they continue spraying well. I also prefer the clear ones, as you can see what the level is inside and orientation of the feed tube as you get towards the end.

... but I don't know how well it would work with coconut oil if you're in a cold climate. At the very least, I would treat it like a can of spray paint, and after using it, turn the container open and hold the spray button down until it's only blowing air. This way, you won't leave (as much) oil in the areas where it'd be difficult to melt it.

I'd also look for one of the sprayers with a filter at the end of it, but that doesn't mean that you could just warm up the bottom of the oil to use it -- you'd need to warm up the sides sufficiently for the air pressure to transfer and you'd need to make sure that the intake tube in the center is warmed up. (and the spraying mechanism, so it doesn't seize as you try to use it).