Here's a simple analog circuit that may do what you want (LTspice simulation below).
Pushing the remote push-button (represented here by S1) applies 9V (green trace) to charge up the capacitor C1 (red trace) through diode D1 and resistor R3.
This turns on the N-MOSFET (M1) when supplies current to the LED (yellow trace).
The diode prevents discharge back through the remote circuit so C1 discharges through R1 until the MOSFET turns off (about 4 minutes in the simulation).
The maximum time achievable is limited by the various component leakage currents (that's why a ceramic capacitor is used for C1, an electrolytic would likely have too much leakage), and those leakages plus component tolerances will cause the actual time to probably vary significantly from the simulation (which should not be a problem in your application).
The circuit is connected directly to the battery, but the circuit only has a very small leakage current when off, so will have no effect on the battery life.
R2 controls the LED current and can be varied, if desired, to adjust its brightness to the desired level.
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