@Pete-S What super special requirement is making the web browser live on the same machine as the server stack?
You should have no issue running a web browser from a machine without a "real" graphics adapter. I have a Fedora desktop running as a VM on my server that only has a 20 year old graphics card.
I've run graphics programs from computers that had no graphics adapter, where the main interface was a serial terminal. Ford's IDEAS and Pro-E 3d design software too, so I'm not talking the low end or easy stuff. Doesn't mean I recommend doing that tho!