Your machine has a 15 AMP charging system, so at high speed, you should have sufficient current. Headlights & fuel solenoid will use around 5-6 AMPs and PTO probably around 3-4? HOWEVER- IF the battery is low in charge, it may "suck" a higher than normal amount and thus drop the voltage to the point that the PTO disengages. I would start off by fully charging the battery and see if that resolves the problem (at least temporarily). IF the battery is 2 or more years old, I'd definitely get it load tested. As a matter of course, you should ensure that the connections to the PTO SWITCH and PTO are good. You should unplug/plug a couple times to "wipe off" any "crud".

Yes, there would be either an adjustment or one must bend linkage parts to do it. The "however" that goes along with that is the speed should be set properly and you will need a tach to do that.

In looking at the wiring diagrams for both the machines (917.276020 & 917.276310), I see no differences between the two except for the Kohler version (020) featuring an operator presence relay where the Briggs version (310) features a more "complex" seat switch to accomplish the same goal. If,........you are connecting the white wire to the small terminal on the starter solenoid, you are making the correct connection,........AND the solenoid is in fact mounted to a grounded, metal surface or has been provided with a separate grounding jumper wire to ground the solenoid case! So, you might want to test that the solenoid is grounded by touching the probe of a test light (or volt meter) to the battery positive post and the other end/lead to the metal case of the solenoid body. If you detect voltage or the light burns, the solenoid is grounded. If you don't detect voltage, it is not grounded and you need to address that first. So let's go through some checks to determine what the trouble might be. The following outline is a step by step that will guide you through. Refer to the wiring diagram for your 917.276020 for the identifying "alpha characters" I refer to in the text. Make certain there is not a blown fuse before you do these tests. (begin test outline) It could be a "bad key switch" but my feeling is "bad clutch/brake switch", but it could be a "bad PTO switch" just as easy, so let's do some testing first. You have a volt meter and/or 12V test light? Be advised that moving or wiggling wires while doing these tests could result in continuity being accidently restored AND this might cause the starter to engage when you don't expect it to. To guard against this, unplug the small white wire from the starter solenoid and cover the wire terminal with tape to prevent a short. First off, you will need to access the backside of the key switch. Step 1. Find the WHITE wire on the back of key switch connector. (leave the connector attached to the switch) While holding the key in the start position, probe the area where the white wire enters the connector body with a test light or a volt meter. You should detect voltage on the test light or the volt meter. If you do not detect voltage, replace the key switch. If you detect voltage, go to step 2. Step 2. Find the PTO switch and identify terminals C & G on the switch. Probe terminal C while the key is in start position. If you detect voltage at C, probe terminal G next. If you detect voltage at C but not at G, replace the PTO switch (make double dang sure the PTO switch is in the OFF position). If you detect voltage at C & G, go to step 3. Step 3. Locate the clutch/brake interlock switch (should be near the pedal or close by, must be attached so pedal linkage can operate the switch. This switch will feature 4 wires, 2 will be white. With the key in the start position, and the pedal locked or held down, probe the white wires of the brake/clutch switch. If you detect voltage on one white wire only, make certain that the switch plunger is actually moving in far enough to operate the switch. If the plunger is moving far enough to operate the switch, but you still detect voltage on just 1 white wire of the switch, replace the clutch/brake switch. The seat switch does not inhibit cranking on this mower, it only functions to kill the ignition spark. Malfunctions in the PTO switch or the clutch/brake switch are capable of killing spark AND inhibiting the starter. Remember to reconnect the white wire to the starter solenoid after testing. I hope this helps in troubleshooting the interlock circuits. (end outline) After doing these tests, post back your results.

The battery should be 12.6V. The fact it only drops 0.2V indicates that the starter motor is not getting current. Make sure all battery cable/connections are cleaned from battery + to cable to solenoid to starter motor and from engine to frame ground to battery - cable to battery - terminal. Since you have a Voltmeter, we can use that to test. Set meter to a range that covers battery voltage. In any of these tests, if you see a multiple VOLT drop, then that connection is bad. A Voltage DROP means the meter reads a voltage. In each test, hook up the meter and then hit the crank position on the switch. A 3rd hand may be convenient (wife?) Keep the RED lead on the battery + side of the circuit. 1. Red on battery POST to cable STUD on solenoid. 2. Solenoid stud to other solenoid stud. (this should indicate if solenoid is bad) 3. 2nd solenoid stud to starter motor stud. If you get a reading on any of these, you may have to isolate the individual connections. Example: In test 1, the connections we have are battery post to battery terminal, to battery cable, to other battery terminal to solenoid stud. Test 1 covers ALL those connections, so if you get a reading, you'll have to go back and test each individual connection. IF no readings on any of those, we go to round 2! Red lead on starter motor housing. 1. Black lead on engine, then frame, then - battery terminal. Same things apply about individual connections as above.