Well..... I have found in my experience that the most expensive and ineffective way to fix a problem is to start throwing parts at it.
Having said that, my suggestion would be to start by checking for alternator output. Keeping in mind that the ignition system and any electrical loads (like warning lights, electric fuel pump, etc.) will draw down the battery while you are mowing. Look for battery voltage to increase from the normal engine off voltage of about 11.5 to 12.5 (depending on state of charge) and see if it rises when the engine is running. It should go up to at least 14 volts. Your solenoid is not a part of the charging system. It is simply a heavy duty switch for switching the high current that the starter requires.