I have never seen any soil test kit worth the money. If you want a comprehensive soil test go to a soil test lab. Basically all you will get with a store bought kit is whether your soil is acidic or alkaline. It will also tell you what is in the soil but it will not tell you if it is available for plant uptake. The DG product is an excellent product. The Sunniland is counterproductive to the DG product and possibly be a detriment to your tree and any possible plants you may plant. It will be much cheaper, efficient and safer to just but large bags of compost, a hard garden rake and a big bag of OMRI rated fertilizer and a grass plug tool. You can probably rent the plug tool. All I ever did regarding plugs is to just use a shovel. Most nurserys will sell St Augustine by the square. You won't need many if you just cut the square into small pieces. St Augustine comes in numerous varieties but most of them require full sun. Here in Texas the best shade tolerant St Augustine variety is Palmetto. Do NOT get Floratam.
Got it, alright, now I just need to figure out where to get a small quantity of compost. There are some locals on Craigslist but they only sell in bulk, like by the truckload. And this isn't a large area of lawn so i'm going to have to do some searching.