I’m not familiar with ODYSSEY but if you can live with it being turned on and off with a relay then turning on that relay with a darlington pair might do the trick. With the base of the transistor(s) being fed via a resistor from the charge on a reasonably large capacitor the relay will be held on for a time determined by the size of the capacitor, the size of the resistor and the gain of the transistor which can be very high. The downside is that the relay coil and the collector emitter junction will be permanently connected across the battery (unless you incorporate an auxiliary switch for the system) but the leakage current will be low.
Happy to clarify further with possible component values if this sounds like an idea…