This is the content of my firmware directory. I was under the impression I had downloaded the correct files:
pi@raspberrypi: $ ls /lib/firmware/*si*.fw -la-rw-r--r-- 1 root root 15827 Sep 15 2016 /lib/firmware/dvb-demod-si2168-b40-01.fw-rw-r--r-- 1 root root 155896 Dec 27 14:51 /lib/firmware/dvb-demod-si2168-d60-01.fw-rw-r--r-- 1 root root 159699 Dec 27 14:59 /lib/firmware/dvb-tuner-si2141-a10-01.fw-rw-r--r-- 1 root root 166427 Dec 27 19:41 /lib/firmware/dvb-tuner-si2158-a20-01.fw
Once I got it, I downloaded the firmware for it on Linux (specifically dvb-demod-si2168-d60-01.fw and dvb-tuner-si2141-a10-01.fw from here if you want to use the same tuner), tested it out, and got way more channels as expected. So then I put the firmware on my Raspberry Pi 3, plugged in the tuner, and went to set it up in my Docker Swarm. But I found a problem with the stack.