Applying foliar applications and soil soaks to lawns wouldn't be too outrageous when we look at the cost to brew a batch of Compost tea. It's about 99% water with a few pounds of high quality compost in a straining bag and some innoculats added to the brew.
Compost extract would be there better route for applications especially if you are looking for a more fungal brew to apply.
Benefits of extract over tea: reduced time, energy,increased nutrient density, and the biology retained during an extract is higher.
When brewing a tea, usually it's Aerated for 24-48 hours, by pumping air into the system you are creating a highly aerobic environment and by default volitizing nutrients amd shifting the enviorment.
With an extract you reduce the oxidative aspect and retain more nutrients and biology in your brew and keep it more fungal (if that's what you are looking to achieve)
Where the tea/extract can increase in value is when proper innoculants are introduced into the brew to get the tea/extract where you want to go for applications.
You can look at your compost under a microscope and easily visualize what's going on, is your compost geared more towards bacterial dominant? (Thermophilic) or is it highly fungal (Johnson Su)?
What food sources are you going to add to feed the biology before application? Fish hydrolosate for a fungal food or a simple sugar source to proliferate the bacteria?
Are you planning a soil soak or foliar application? For a soil soak, incorporating water soluble AMF powder would be a good addition to colonize with the rhizosphere. If you are doing a foliar application the AMF wouldn't necessarily be a key beneficial, you would incorporate something more along the lines of EM for the endophytic biology such as saccharomyces cerevisiae or Lactic Acid Bacteria. (Which are easy and cheap to brew).
Carbon credit generation takes some time and/or energy. One would have to create 1T of biochar to get the 1t credit during applications. Seems simple enough, but still takes energy and time.
Another route would be focusing on increasing Soill Organic Matter, season after season through cover cropping and green mulching. By pairing with AMF to increase the carbon sequestered into the soil (increasing root surface area as well for better nutrient retention, water holding capacity and pH buffering).
This has to be tracked through a soil sample where you first get your base level of Soil carbon tested and recorded, then you can record any increases year over year. The healthier and more robust the rhizosphere, the more carbon sequestered and higher credit generation over time.
