I have replaced a lot of LCD panels and full lid assemblies for this model with no issues, so it will not be a screen problem,
Sounds like the keyboard may be shorting PP3V3_G3H or PP5V_S0 to ground and preventing initial boot when it is plugged in. These are the only voltages which could be affected by the keyboard, the rest are trackpad related. Nothing links to the battery itself as far as I am aware, but this model is a lot different than older models (for example, will not boot after disassembling unless you plug the charger in), so it may show the no battery screen if certain signals aren't present.
I would buy a test keyboard and go from there. Check the whole of the logic board for liquid damage too, this model is badly designed, so the liquid tends to pool around the board, since there is nowhere for it to drain.