I think a 36V throttle will work. The battery charge indicators will definitely not work (it will show a dead battery). I have not tried it myself. I have gone the other way and run a 24V throttle at 36V. I think the hall sensor part of the throttle (the part that tells the controller the throttle position) is likely to be identical. I expect that the only difference is some resistor values in the circuit to light up the LEDs.
One way to make sure that the throttle is the problem is to put the bike up on a stand so the rear wheel is free to spin and then pull the throttle connector. There will be five wires and the colors vary by year. One is positive from the battery and one is ground. The remaining three are the three hall sensor phases. With the bike on, shorting some combinations on these three should make the wheel spin like full throttle. If this happens then the throttle is indeed the problem. Note that you need to take your time with this and make sure that you have really shorted the wires. I mislead myself into a wasted hour or so of troubleshooting when I wasn't really making contact.
The throttle is the most likely cause of "won't run" issues on these bikes followed by the relay so a simpler way is to just throw a throttle on there and try it. As long as I am here:
My personal Oset troubleshooting steps for "won't run" (I like to do them in order of easy and likely):
1. Make sure batteries are charged. To totally eliminate this as an issue I check it with the charger on.
2. Make sure I hear the click when the key is switched on (if not then it is likely to be the relay or relay wiring).
3. Double check that the potentiomter is open enough. The pots can fail so disconnecting it and shorting the wires will eliminate this.
4. Check the throttle with the shorting test above.
5. If none of the above get you there then you might have one of the rare complicated issues. You can troubleshoot the wiring if you have the skill or contact your dealer if you don't.
Way more answer than you asked for, but I thought I should write it down for others.