There are a number of threads/posts about this sort of thing; in the cases where anything is known about the displays, they have a very tiny transistor capable of sinking only a few mA (a few dozen mA at best) that you might be able to use to drive another larger transistor with that can then be used to turn the light on or off with.
I don't know what the rating of the light output is on those; if it doesn't state what they are then it's safest to assume they are very very low capability sinking (grounding, rather than sourcing, or supplying voltage).
If you don't know how to design a transistor (or FET, relay, etc) switch, you can buy modules that already have (most of) the stuff necessary on them. Assuming the display's transistor is used in a typical way as "open collector / open drain", it just grounds its output when turned on, and is open circuit when off. You'd install a 5kohm-10kohm resistor from that output to 5v, to create an on/off signal that's ground when on and 5v when off, then use a module with an input that can take that signal and make it's output also ground when on, and open circuit when off. The module's input gets connected to the signal, and grounds shared. Then the light's ground connects to the module's output, and the light's voltage input connects to your battery supply.
If you can find a low-enough coil-current relay module you can use those instead, sort of like these:
https://www.amazon.com/Tolako-Arduino-Indicator-Channel-Official/dp/B00VRUAHLE
but that specific module might not work, because according to this page
https://www.circuitbasics.com/wp-content/uploads/2015/11/SRD-05VDC-SL-C-Datasheet.pdf
it requires up to 90mA at 5v to turn on the relay coil; your display may not be able to source or sink that much current.
Might need one that has it's own onboard relay driver that is powered from the main battery voltage instead of 5v (so you don't overload the display's or controller's 5v and damage something), and takes an "active low" input.
