If it's a small load like a timer motor, the appliance could be "hacked" by adding a tiny 240/120v transformer to produce the 120V within the appliance.
I would consult with the factory to see exactly how they use 120V within the appliance and if they have any sanctioned kit or instructions to do just that.
Very often, 240V appliances have some control that is 120V. When they sell that same appliance in East Asia, which is all 240V-only, they add a tiny 120V transformer just as I describe. That "East Asian" spec is perfectly compatible with US power and they could market them here... but they want to pinch pennies, so they remove the $2 transformer and require a neutral.
So if it's possible to convert the appliance to East Asian spec, that may be cheaper than running new cable with neutral.