Easy peasy if you & only you use the mower
You will need a starting solenoid and a starter button which can be any spring loaded switch, like a horn button for example
Battery + to switch to solenoid trigger terminal .
This is of course UNSAFE but if only you use the mower and you always sit in the seat before you start it then you won't accidentially cause the blades to rotate nwhile your bare toes are under the deck ( not pretty )
Rectifier + wire to the battery terminal on the solenoid & you are done
Existing switch should work fine to stop the engine