This could be fun.
To make your life easier, I suggest an 8S pack so that the pack voltage is > 20 V. That way you can use buck converters, which are cheaper, smaller and more efficienct than buck-boost.
Analogue control might work quite nicely. A decent synchronous buck converter with an output voltage tracking feature might work for the 5-20V output.
For digital control, I would suggest a microcontroller rather than an FPGA. Writing firmware is still a pain, but much easier than writing VHDL / Verilog. To select a chip, peripherals are everything: you need good PWM (high clock frequency, hot/cold outputs and dead time) and a good ADC (say 12b 1MegSample or higher) and a hardware synchronisation line from the PWM module to the ADC.
The Texas Instruments C2000 piccolo chips are quite good (good PWM, high speed ADC) for power electronics. Alternatively, I think there are some ARM chips which could fit the bill, but haven’t looked into them.