The purple wire on the 160 should be the trigger wire to the starter solenoid and should go battery voltage when the key is in the Start position
So it goes to the trigger post on the remote starter solenoid for the Briggs engine the relay YF4-5511-SO1 is now surplus.
If you look at the full circuit diagram you originally posted you will see the purple wire ultimately ends up at the S on the key switch after going through all of the safety switches so the engine can not crank unless the PTO is off and the brake is on .
The Ignitor on the 160 is also redundant because the ignitor chip is embedded within the coil on the Briggs engine
Thus the green wire will connect to the back wire on the Briggs engine.
The unknown do dad near the battery is a solenoid switch .
If it interrupts the battery starter cable, bypass it .,
Sorry for the delay... work and such.
Reading your response I can't help but question if you have the engine swap going in reverse? The Kawasaki from the 160 is the good engine, and going into the LA110 with the blown B&S engine. My questioning is because:
1) You state that the purple now goes to the starter on the Briggs and the relay is of no use.
2) ... the ignitor chip is embedded within the coil of the Briggs.
What I see so far is that
1) The thing I notated next to the battery in the LA110 diagram is actually controlling the starter so when the key is turned, it clicks loudly and sends 12v to the starter. SO... I guess I don't need the relay or the purple wire to connect to the 160 starter, I can just run this power to it.
2) Next thing is the alternator. The 160 uses a voltage regulator, so I assume I must wire that into the 110. I can pull the power off of the starter solenoid, disregard the yellow low fuel lamp wire, black to ground, red/ whites stay going to the alternator, green goes to ..... ??? hhhmmm... it states it connects to ground at the switch when in off position. When I look at the 6pin on the LA110, I have red/ black with constant 12v, yellow with 12v at key "on", yellow/ white and white. If I had to make a guess I would say green to white as the white on the 110 originally connected to the coil... but that is a weak guess.