What are the total current required to run all that hardware at the same time ?
Even if you make it run it will not run for long with only one 18650 battery.
How long time do you expect to run it before replacing the battery ?

Think you need to share the code you have loaded on the esp32 else we cant see what goes wrong, just my 2 cent

I dont have any ESP3-S3-BOX-LITE so cant tell.

Bluetooth use less power than WiFi, so it should be possible. If you measure the current used, you can calculate the required for the batteries. Suggest you try to run it with a power bank and see how long time it will run. Just make sure the power bank dont shutdown when low current is required.

What are the requirement, how long should it run on the battery ? does it use wifi ? Bluetooth ? Do you know how many milli Amp it require ?

Im not using Micro Python, just trying to assist you. Are you following freenove or esp own guide and bin file ? I think you need a version with the correct size of psram and flash ?
I have some freenove ESP32-S3 modules and they work great with Arduino ide

It seems you Python code make the esp32 reboot, try with some minimum code and if you still have an issue, then post your code here.

You can build a web server running on the esp32 and then control everything from the web page, you can find lots of guides on how to do it, and then connect your iPhone to the webpage, i know its not an App and not bluetooth, but it works

If its Arduino nano Esp32 (was released some time ago now) i think your best choice is the Arduino forum, if its another board please supply link to it ?