Ok…
The GW spray gun isn’t a very good airbrush because it isn’t and airbrush it’s a spray gun. GW never marketed it as an airbrush but the paint+air=painted stuff equation made a lot of people think it was intended as an airbrush. That said it is(was?) a great spray gun that does exactly what it says on the tin that was easy to use and maintain.
For the description of what you are saying you want to do a spray gun would be fine.
That said an airbrush would give you a lot more options in the long run, just because you only want to base coat this week doesn’t mean you only want to base coat forever.
So assuming you go with the airbrush option there’s a few things to consider:
Single/Dual Action.Single action allows only for an adjustment of air flow, dual action allows you to adjust the air flow and the paint flow. Dual gives you a lot more control, but may cost a little more as can be a little harder to maintain.
Gravity/Siphon feed.Gravity feeds generally have a small cup mounted on the top/side of the airbrush and use (drum roll) gravity to help feed the paint into the airbrush. Siphon feeds normally have a glass bottle that plugs into the bottom of the airbrush and uses the air flow to draw paint up into the brush. Gravity feeds tend to be a bit easier to work with, easier to switch colours with and less wasteful. Siphon feeds allow you to set them up with more paint so you can spray for longer without having to keep topping up.
Internal/External mix.Internals mix the paint and air inside the body of the airbrush, externals mix the paint with the air flow after the air leaves the nozzle. Externals are a little easier to maintain and… well have nothing else going for them, 90%+ of airbrushes you look at will be internals.
For the record the GW “spray gun” would class as a single action (Pull trigger to spray) siphon feed (Glass bottle under the brush) external mix (The paint chamber is mounted in front of the air nozzle) airbrush… Kind of…
Last thing to consider is the air supply. Compressed air cans are cheap to get started with, but by the time you’ve done 5+ cans you could have bought a cheap(ish) compressor, they also run out at annoying times leaving you unable to do anything until you get another, which depending on where you live can be a major task. Cans also suffer with pressure drop, when you spray the cans temperature drops drastically and so does the pressure, also towards the end of the cans life you’ll be working with a greatly reduced pressure from what you started with.
Compressors are more expensive to start with, take up more room and are noisy. But the are cheaper long term, give a more consistent (And often adjustable) air pressure (Doubly so if the compressor also has and air tank, a feature that also helps reduce the noise as the pump will shut off after the tank reaches a set pressure and only start again when it drops a little.)
In short a single action, siphon feed airbrush would cover you for what you currently plan on doing (base coating). If you wanted to think more long term there are
some good sets like this that would give you a lot more options (Couldn’t find it on the US Amazon, but £88 works out at around $140, just under your top end budget but most of that money is going on the compressor)
Do some googleing (The terms I used above should help you get started) and check out some youtube videos, there’s a lot of info out there and kit that ranged from $20-$1000+. Also check out price/availability of compressed air cans.
*Edit*
Beware top end brand named airbrushes at crazy low prices, if it seems too good to be true and is shipping from China chances are good it’s a knock off, while they can still be perfectly good brushes in their own rights you’re not getting an amazing bargain just a cheap brush with another companies name stamped on it.