Compost in the form of tea or top dressed will both work equally well. Tea will get to work faster, but unless you have some sort of huge brewing area, that kind of lawn is going to take a lot of tea...Typically it's brewed in a 5 gallon bucket with fish pump for aeration. You would need like 50 gallons of tea to cover that lawn.

I have about 5000sq feet of lawn and its easy enough just to load up the wheel barrel and scoop shovel fulls of compost and sort of just heave it across the yard.. doesn't need to be too uniform, it will be spread out by nature rather quickly. This method won't work for you, I have never used this tool, but you could consider buying a compost barrel spreader. It's a relativley cheap option, since the next step up would be some sort of commerial tools

Compost Barrel Spreader

Compost tea is one way to get some bacteria into the soil, but if there is not adequate amounts of organic matter in that soil they will have nothing to live on and will not be of much use.
If the soil has adequate amounts of organic matter in the soil spraying compost tea probably is not necessary.

Thanks for the information! Really glad I found this forum. I just ordered Paul Tukey's Organic Lawn Care Manual and look forward to diving in.