If the battery isn't charging and I believe you said you had changed it out for a new battery then you need to have the Stator checked or replaced. It regulates the voltage that the battery receives, and if the Stator isn't working right your battery will never hold a charge. The Mower will run but only as long as the battery holds up and if you have an electric PTO that's where you'll notice the failure first. Do have a question though, what type of mower are we talking about??
The OP posted that he was reading 14.3V at the battery when the motor was running, which would indicate the charging system is working (he didn't state the engine RPMs when testing). He hasn't posted since he swapped out the battery.
The stator generates AC power through a long coil of wire passing by magnets, and the wattage output varies with engine RPM (higher RPM= more power). The AC power is carried to a rectifier that converts AC to DC, then to a voltage regulator. Most times the rectifier and regulator are in one package. Many times the electric lights are powered by AC power directly without conversion to DC or any regulation. If you measure the AC as it enters the rectifier/regulator you should see 30V+ AC.
If you have 30V+ at the rectifier/regulator you should have 13.5+- VDC output, if not, replace the rectifier/regulator. If no 30+ VAC, then check the stator resistance end to end, and to ground. You should have an open circuit between the stator coil and ground, and a low resistance measured between the two wires from the stator. Low resistance would be close to 1-2 ohms; an open would indicate a broken stator wire; a high resistance might be a bad connection. The magnets that are used to excite the stator coil are prone to coming loose from their mounts, and when loose may trash the stator If the stator coil isn't shorted to ground or open end to end, yet there isn't any AC power, check the magnets.
Good information on B&S charging systems can be found at
http://www4.briggsandstratton.com/miscpdfs/RNT/alternator_replacement.pdf . Note that B&S refers to the stator, rectifier, and regulator as an alternator.