If it's not actively being used, power consumption shouldn't be too high, even with GPS and data turned on. (Do flip phones have tracking? I admit I don't know - but if they do, just get one of those. Mine only needs to be charged a couple times a week.) That said, it seems to me there are a couple ways around that.
If it's more likely to be stolen at night, just charge it up during the day and plant it in the evening. Alternatively, keep a couple external power banks in rotation.
If the display you're protecting runs off wall power, you could use that to power the phone as well. The simplest way is just to plug it in at the same source and hope the thief doesn't think about it too much. If you want to get a little sneakier (and don't mind voiding a few warranties), you could wire a USB adapter into the display itself.