One trick might be to asap put the onboard vga card into MDA mode,
then initialize the PCI/etc card. If the onboard card does a good job of strictly emulating MDA, coexistence with another vga card should be no problem (port and memory IO wise).
If you can pre-initialize the onboard card into MDA mode (before the
system bios does its normal check/vga initialization), the bios *might*
try to do the right thing and initialize the pci card normally.
This involves calling the onboard vga bios first, setting MDA mode up (by a hack), then hoping the bios detects/calls your pci card's video bios.
It's weird that your system can run without video. I would guess some int10 modes like "get the cursor position" or "get video mode info" would get confused and cause programs to crash.