There are some questions that need answers before you start, especially, how long you intend to have it operating on battery power before recharging. For some reason, your first link does not respond for me so I can't speak to the display but the controller board will require 12 volts at an idle power of 0.5 watts. The RPi 4 has a recommended power requirement of 5V at 2.5 amps but I have heard of them drawing much less power. so the range is wide but the voltage requirement is strict and the "official" supply is spec'd for an average user's power consumption with some headroom. It will require a REGULATED 5V for the RPi. In any case, you must provide for at least two voltage levels and depending on the display, possibly more.
LiPo's are safe enough as long as you adhere to the charging/discharging requirements strictly and prevent overheating. It is best to go with the more conservative ratings on these cells and also to avoid the super-compressed form factors used in some smart-phones. They have a nominal voltage of 3.7 V but fully charged are up to 4.2. Full discharge is at around 3.2 V. Charge rate recommendations are all over the map but I tend not to believe the higher claims of some. Panasonic is a pretty good reference for conservative ratings on their various cell models. Other outfits may actually put their own packaging on cheaper cells and give outrageous numbers for charge/discharge/rate ratings. The display requirement for 12V means either a stack of cells or a boost regulator, or a 12 V battery.
You need to decide your runtime needs and the current requirements at each voltage level along with the weight and size requirements you may impose on the system. Add 10-20% at least for headroom. Then you can make a decision on battery/regulator(s) if any/charging methods, etc.
My basic advice is to go with one of the larger RPi HDMI displays and use separate supplies for the RPi and display (with a common ground)...forget the bigger display and controller.
As for the Adafruit 1000C...fine product if your current requirement does not exceed 1 amp but the RPi4 will require twice or more that current. As for larger boost regulators, you are on your own here but a good search engine is your best bet for info. Again, discount wild claims of current capacity, etc.