LCD controller works with adapter but not with batteries
Pretty elementary: your boost converter (Datasheet: http://www.sunrom.com/get/283300) can't supply enough power (not even close).

The IC is rated at 4A absolute maximum input current. At depletion your Li-Ion cells will be around 3v. 3V*4A = 12W (theoretical). Even at 100% efficiency this equates to just 1A of available current at 12V, so you need to juice your power delivery capability by 300% (and then again my another 20% over that (minimum) since you can't run PS components at redline and expect things not to mushroom cloud. 

My advice ?

Ditch the 1S4P battery design, get one of those cheap car starters off Amazon that include MicroUSB charging. Use the 3S LiPo battery, inside it and add your 3Ah cells to it (having one left over) for a 3S2P config and ditch the boost converters entirely, using the 2A USB buck converter in the pack for the RPi and run the peripheral board direct from the cells. 

https://www.amazon.com/gp/aw/d/B0191DLAZ0/ref=mp_s_a_1_4?ie=UTF8&amp;qid=1492167596&amp;sr=8-4&amp;pi=AC_SX236_SY340_QL65&amp;keywords=bestek+jump+start&amp;dpPl=1&amp;dpID=41YAstgOlZL&amp;ref=plSrch

Have fun on your journey to realizing just how cheap laptops are ! ;)

-=dave