I have an LS1028 which I’ve not yet tested. It’s the next module for which I want to get NixOS working, so I’ll get that installed, hopefully later today, and report back if you haven’t resolved your issue by then.
Edit: I’ve got the LS1028A module installed and booting. Works fine on my box, sorry to say. I’m running the 3903 image as well, which I found linked on the Reform documentation page. (I also tried to use the link in the handbook at first. Seems the /system-image redirect needs to be updated.)
My LPC and keyboard are both running the 2021-04-19 firmware.
I did notice that I had to disconnect the keyboard from the motherboard in order to comfortably install the eDP adapter, and the system does boot with it disconnected, though it (as one might expect) doesn’t accept keyboard input once booted. It might be a dumb question, but you do have that cable firmly connected, right?
Aside from that, my best guess at this point is no better than yours: try upgrading your firmware.
Edit 2: Might also be the physical connection from your keyboard to the module, aside from the connection at the jumper. I had a physical trace broken on my keyboard when I got it; was able to bodge around it, but still, it does apparently happen. Wouldn’t explain your serial troubles, granted. Have you tried connecting a USB keyboard to see if that works?