Only after the code is working one can know how much computing power is required for this job.
The choice of the operating voltage affects all connected sensors and actuators. Since it's easier to get more powerful controllers for 3.3V than for 5V, I'd start with the 3.3V version.
But instead of the Pro Mini I'd choose a board with wireless connectivity for remote control.
The 3.3V 8MHz Pro Mini will work fine with a LIPO battery connected directly to Vcc. In fact, you can remove the voltage regulator and power-on LED and save a bit of current draw. It will also work with 5V connected to Vcc, but will still run at 8MHz.
You might need the 16MHz version if the 8MHz just isn't fast enough. If you need to connect to external sensors or devices at 5V, you could drive your 8MHz Mini at 5V. But you shouldn't connect the 3.7V LIPO to the Vcc of a 16MHz Mini. The Atmega needs 5V to run reliably at that speed. So if you switch to a 16MHz Mini to get the extra speed, or if you run the 8MHz at 5V, you'll have to use a different power supply such as adding a boost regulator to the single LIPO.