Right away i can see you have no oscillator for your arduino. I suggest looking at the official Arduino website for the schematics of an arduino Uno (whitch uses the Atmega328P-PU) and copying as much as possible.
https://www.arduino.cc/en/uploads/Main/Arduino_Uno_Rev3-schematic.pdfYou will also need a programmer (externall, onboard, or another arduino) (the programmer is included on the above schematic).
GreatScott has made a very nice video about Standalone Arduino Circuits where he explains how to build one, and how to program it.
As far as the battery goes, i am no expert. In my experience however Li-po batteries are easier to work with. (18650 batteries are a prime example)
They have a plethora of BMS modules you can choose from, and are incorporated into many designs which you can find and follow.
GreatScott has again two videos regarding BMSs that are both great. (spoiler : dont build one, buy one)
Also i see no power supply in your design. You should use an SMPS (No Post-SMPS LDO required) aimed toward battery-powered applications, they are not hard to find. The topology of the SMPS is dependent on your battery's voltage.
I do not see any other problem at this time. Overall your idea is sound. But in my experience the procces of integrating arduino modules on a single PCB is not always cost effective . I would personally buy the external modules, put them all together, and then hide as much as possible in a box. But this is your choice. Regardless your idea is great, and i might even follow it in the future with some spare nixie tubes i have lying around.
Good luck in your efforts.