Microcontroller vs FPGA to DAC and ADC
As I see it, you have 3 options:

PC, scaled down

Keep using the same system, just scale it down. This is easy (because you have a high-level system), but maybe not too efficient since the operating system is not optimised for latency or real-time operation. It will also be power-hungry.

MCU

A microcontroller running at ~100 MHz ought to have plenty of processing power for this task (and will be far more responsive than a linux based solution). You will need to interface to the peripheral voltage levels, but should be able to use a simple amplifier for the task (providing some protection for the digital side in the process). You can capture output data over a UART, some indicator lights or a small display panel - all of these parts should be easy to find examples for. Of course, you have to learn a slightly different software paradigm (micro-python is possible, but maybe not ideal here).

FPGA

Your example part might not be a good fit - it is hugely over-specified and expensive. The basic real-time processing could likely be done in a fairly small device, but the low level coding is likely to be a challenge (unless matlab can generate you all the code you need).

Using FPGA here would typically be the approach to use if the calculations are complex and demanding. An MCU ought to be a good fit if your aim is to scale down the hardware, and maybe make it battery powered. Plenty of small MCUs come on credit-card sized development boards at ~$10.

Check what data rate you need in order to generate your 'canceling' signal, this will determine the memory array size and DMA frequency that you need to drive the DAC in your design.