The ATMEGA328P is not the best IC to do ADC measurements with, plus it is very out of stock.

I'm thinking for the final design using a Laser Bee from Silicon Labs might do the trick. It is designed specially for measuring analog signals and it is quite a low cost part. It also has USART to pass messages to the BLE module, and it wouldn't need the finagling that the ATMEGA needs to run at 3.3V.