The treatment with iron sulphate is not critical. If you do not want to make the soil more acid you can antagonise that with liming (50-100 grams/m²).
At any rate, it is possible that the moss carries weed seeds within. So it depends on how weed-rich the lawn, from which you removed the moss, is - in order to be able to decide whether to use it or not.
Compared to bark mulch the moss lies lose on the soil. To encourage the rotting it would be good to incorporate the moss somewhat into the ground and, depending on the plants, use lime.