How fast the grass disappears will depend on your weather, time of year, occupational density, age of the chickens, the type of grass, type of soil, and how well the turf is established. You’ll just have to see what number of chickens is right for your unique situation. Where I am that varies by season.
You have a bit of a challenge to meet your budget. Don’t forget foundation costs. Craigslist and Habitat Reuse stores are great resources. If you buy building materials, most come in 4’ or 8’ lengths so you can usually reduce cutting and waste by using those dimensions but that depends on what material you are using.
In your climate I’d build an open ended coop. Make one wall out of wire. Create a cul de sac to put the roosts in so they are out of direct wind, put a roof over it to keep out rain, and make as much open as you can. Heat is your enemy down there, not cold. They could sleep in trees down there and do fine as far as the weather. I’ve seen chickens sleep in trees in zero degree Fahrenheit overnight lows and they did fine. In your climate, cold is not an issue.
You could even make the whole thing open except for wind breaks around the roosts. I’m thinking more of your hurricane winds or afternoon thunderstorms than the cold.
One problem with an open air coop is that rain blows in. A wet coop or run is a dangerous coop or run, a breeding ground for disease. A wet coop or run also stinks. As long as that floor is higher than other areas so water has a place to drain to and you have good ventilation that should not be a problem. If the floor is sand so it drains well, that is even better, but the water needs a place to drain to. Good ventilation and good drainage should dry it out pretty quickly. With that many chickens poop will build up, especially under the roosts. I’d give a lot of consideration to a droppings board to reduce the amount of poop building up in your floor. It’s quite likely you could get by with no bedding material, just the ground, especially if it is sand.
That’s another reason to build big. The more you can spread out the poop the less hard you have to work to manage it. If it is concentrated in one place, like under the roosts but also in small coops or runs, it can be a problem.
You might follow the link in my signature below to get my thoughts on room. I don’t give you any cut and dry space recommendations, more of things to consider when you are planning your size.
Good luck. What you are talking about should work but you need to be flexible enough to adjust to your unique conditions.