Just keep in mind that something like stall dry is going to change the ph of your yard. They aren't meant for use in a 'garden' area and since stall dry contains diotomaceaus earth (not spelled correctly) it's also going to kill any insects in it's path. While that may seem good you have to remember that some bugs are good
It's also got clay and you are expecting rain ...
Plus DE is not good when inhaled so you'll have to wait for it to settle. Not poo pooing it. I think spirit just uses it in one spot and it sounds like you want something for the whole yard which makes it more difficult.
Can you hose down the pee spots?