You may have two differents problems related to your issue...
1) you have a poor battery and maybe poorly charged so you cannot expect much of it.
2) the power efficiency of your screen is really bad because of the convertirs thus you cannot last long while activating the LCD and that’s why you have a better battery life when playing on the TV.
3) this is an extra, adding an external controller to the build via bluetooth of whatever will consume more power also
Take a look at my signature, i have a special
Post for optimizing the system for battery and SD Card life. Also i’ve made some testings with my system, and my 4800mAh battery gives me 7,5h of intensive playing before dying (on LCD).
So what i suggest, test a different battery even if you let it outside (just for a test, like a branded mobile phone battery)
Do not hesitate to solder the wires to the battery pads directly and run on it, you will be fixed if for example, you get 2h with a 1600mAh cell phone battery.
As for the power consumption of the LCd driver, i have nothing to say to help you except if it’s really the only issue, change screen
I suppose this a BW composite screen ?