It's true that most FPGAs have limited built-in analog capabilities. But good DACs and ADCs aren't too expensive, and an FPGA can control them with exceptional precision. Does your process have some kind of input / output that can't be handled that way?
Dedicated DACs/ADCs will almost always offer better performance than the ones you'd find on a microcontroller or even an ASIC.
You don't even need DACs or ADCs, you just need an SRAM cell leaking current through an electrode. The process is entirely separate from that. By analog, I don't mean the signal, I mean the chip is interacting with the world physically. I need the electrodes physically touching chemicals. Those chemicals are incompatible with the aluminum or other normal metals on those connections - you pretty much need platinum or the metal from the chip will screw up the chemical reaction.
You also need large amount of input/output - a good start on a chip would be about 1,000 to 10,000 electrodes. I think it is going to be difficult to put that many on an FPGA.
Dedicated DACs/ADCs will almost always offer better performance than the ones you'd find on a microcontroller or even an ASIC.