Snapper lawn tractor repair?

snapper lawn tractor,LT145H38DBV, Briggs and Stratton eng, Ihave replaced the starter earlier this year and a couple of weeks ago I shut it off and tried to restart the next day and nothing,slight click from the solenoid but that's it.I replaced the solenoid with a new one(Trombetta,which bought out the OEM manufacturer) and still nothing.I had the battery load tested and proved to be junk,so that was replaced,and yet nothing.If I jump start it directly to the starter,it will start,but only runs for a few seconds because the anti-fire device on the carb,it's not getting power.I was told by the parts place that the new solenoid has to get a second ground to the flag connector on the solenoid due to a re-design,tried it to no avail.I have worked in the auto collision industry for 40+ years,so mechanical apptitude is not an issue when responding.I do not,however,have at my disposal electrical test equipment to bench test components.My next step is the old "Louisville Slugger" method of repair.I've hit the wall with this thing,can anyone help?