A relay and bulb will be fairly power-hungry compared to a LED and electronic latching circuit. That means the gadget will need a much larger battery, as the relay coil will be on all the time the bulb is. Consider using the simplest electronic latching device, a
Thyristor aka: SCR (Silicon Controlled Rectifier) for the latching circuit as they hold on while load current is flowing, without needing any coil current. Just about any
TO-92 package SCR would do the job, e.g. the
NTE540x series should be readily available in North America, e.g.
https://www.alliedelec.com/product/nte-electronics-inc-/nte5400/70215202/ or can be found on Amazon.com. You'll also need a few resistors, one to limit the gate current, another as a gate pull-down to prevent electrical noise causing false turnons, and one to limit the LED current. Ask if you need help with the circuit.
One cute addition, assuming a front opening dishwasher with a door that hinges down, would be a tilt switch to auto-reset the circuit when the dishwasher is opened.
IDEngineer's idea is good, but rather than relying on hysteresis to hold the state, it would be better to use a sprung popup 'flag' released by the bimetallic strip. You could probably hack a mechanical hot-water cylinder external (strap_on) thermostat, and simply pinch a thin tin-plate or aluminum sheet flag cut from a flat sided can between the closed when cold contacts. Bevel the contact edges with a fine file and 'break' the corners of the length of that edge of the flag so the flag can be easily pushed back between them to reset it.