Some updates:

1. I got coreboot and SeaBIOS working on the machine with the V4 MXM card as the primary card. It's a long story, but basically I moved the VGA ROM execution from coreboot to SeaBIOS and axed the iGPU so that it isn't detected by SeaBIOS (which was way harder than I thought it would be) during init.
Installing XP fails with "System is not fully ACPI compliant". I found a few ACPI related issues and fixed them, but it still fails with that error.
98 freezes shorty during startup.
DOS and Linux work fine.

2. Managed to finally assemble a second card/laptop. Found a bug that wasn't present on the first card/laptop, most likely due to silicon variation and whatnot. Fixed that.
I'll probably assemble 3-4 more laptops/cards, just to be sure that everything is fine.

3. Found some FIFO related issues on the FPGA side and fixed them.

4. 1280x960 now works thanks to the timings provided by @3dfx_fan here: Lost Joker 2 - image shifted to the right

I'll start assembling all the documentation/files for this project, and if the next few laptops/cards work fine, I'll publish everything. Haven't tested the latest FPGA code on the eDP variant, but it likely still has issues.

Also made this in the meantime, currently testing it, but it seems to work fine:

Always lovely to see the results of your work sdz. Quick question from my end, how many layers do you use on your projects. I suppose this MXM one might use more due to the density?

@tehsiggi Thank you!
You are right, due to the density of components, and the extra components so that this can work in an MXM slot, it has quite a few layers. The V3/V4/V5 MXM cards all use 12 layers.

Edit: For the V5, I would have used more, since there's the SLI/AA bus between the two VSAs, as well as the PCI bus, but JLCPCB can't do more than 12 layers @1.2mm thickness. It was a bit of a pain to route it.

12 layers, that is quite hefty!

Given that the surface basically is covered with components, a logical choice.
Again very nice work. Looking forward to further updates!

I managed to get XP working on the machine with coreboot/SeaBIOS and the V4 M4800 as the primary card:
-made a few more coreboot ACPI related patches
-used a newer XP acpi.sys
-modified the V4 M4800 so that IDSEL is connected to PCI AD16 instead of AD17. It was previously connected on AD17, because with the original BIOS and XP installed, the card would not work on AD16, resource conflict error.
All of the above were needed to get rid of the "non ACPI compliant system" error.

At the moment, XP, DOS, Linux work.
Windows 98 is still very much broken.