Fortunately, most procedures have been documented by Netgate. I'm thinking about writing a detailed walkthrough at some point, for now I'll give you the most important steps:
Preparation:
- Download a backup of your configuration ASAP.
- Create an installation USB. Refer to https://docs.netgate.com/pfsense/en/latest/solutions/netgate-6100/reinstall-pfsense.html to to obtain the .iso file. Use a tool like Rufus, USBImager or Etcher to create a bootable USB.
  - Note: They mention support for putting the config xml file on the USB to automatically restore your configuration after installation, but I did not succeed here and had to restore manually after installing.
- Make sure your PC can communicate to the router via the USB console. Refer to https://docs.netgate.com/pfsense/en/latest/solutions/netgate-6100/connect-to-console.html (in my case, I made it work using PuTTY).
- Purchase a compatible M.2 SSD; compatible means:
  1. It must be B-keyed or B+M-keyed, not M-keyed like most SSDs are.
  2. It must be an PCIe/NVMe SSD, not SATA or similar.
  I've purchased and used this one successfully: Transcend TS128GMTE452T
- You will need a T6, T10 and PH1 screwdriver.
- If your WAN connection is not DHCP (e.g. static or PPPoE) write down the information required to configure it during installation.
SSD installation:
- Halt pfSense before removing power (like you should always do).
- On the bottom, remove the indicated T6 & T10 screws: https://imgur.com/a/g0AccMZ
- Remove the plastic bottom plate.
- Carefully pull the indicated sides out a bit to allow the board to be lifted out: https://imgur.com/a/kkq01S8
- Flip the removed PCB over and install the SSD in one of the available M.2 slots using the PH1 screwdriver (note: the third slot is mSATA, which is physically incompatible). It shouldn't matter which slot.
- Either re-assemble the unit or install pfSense with the unit baren (keep clear of objects near the PCB, proceed at own risk).
pfSense reinstall:
- Make sure your router has a working WAN connection plugged in, you can also connect the LAN side to your PC while you're at it, but this is not required yet for installation.
- Before powering up, make sure the USB console is connected and listening as per instructions above.
- Insert the installation USB into one of the available ports.
- Power up device. Refer to https://docs.netgate.com/pfsense/en/latest/solutions/netgate-6100/reinstall-pfsense.html for general installation steps.
- Before setup starts, it will ask you what terminal emulator to use, you can accept the default. Note that the setup can be quite slow so a little patience is required.
- Make sure to configure your WAN so installer can reach out to Netgate and download packages.
- Next, configure the file system and target system. You want to pick nda, not mmcsd. If no disk is showing except mmcsd, power off the device and check SSD installation (maybe it needs to be flipped over, or try the other slot).
- At installation end, set a temporary password for the built-in admin accounts. The backup restoration will revert this to whatever you had.
- After installation complete, connect your PC to LAN and log in to the router using the admin credentials.
- Restore your backup and reboot, it should work like before after all packages have completed installing.