Sounds to me like you will have to choose your power pack carefully and / or make some choices re: the light source to allow continuous lighting while using a battery pack as a power source. Some of the more efficient LEDs are perfectly fine as a light source down to 1mA or so. A comparator would do the trick, and NTC varistors other than the TMP36 may also be cheaper and good enough for your application based on a +/-5*C spec . I'd consider a lower battery voltage (i.e. 2x 1.5V nominal) since that would limit the drop-down resistor needed to illuminate the LED.