My first preference would be to repair the original motherboard. The second would be to find a replacement board, Intel desktop boards are relatively common, at least here in the US. As a last resort, I would get a contemporary board and desolder the ports that don't fit, you wouldn't need most of them except for a USB and maybe an Ethernet port. I like having mine on my home network so I can move screenshots around.
I would advise against upgrading to a more recent motherboard. First of all, there is little benefit because you'd have to keep the old slow W2k. I'm told WinXP can also work but with possible GPIB-related blue screens. The "firmware" has some questionable things regarding timing and concurrency, so its reliability with a faster/multi-core CPU is uncertain. I recall reports of crashes if Windows boots too fast and the vxworks board can't keep up.
Secondly, on non-B TDS5000s there may be BIOS incompatibilities with the PCI interface card, the system wouldn't boot at all. Your pictures show later B-series scopes and the socket 478 implies the same, but I'm pointing it out just in case. Edit: isn't this the TDS5054 s/n B020xxx seen recently on eBay that was smashed with a hammer?
If so, it's an early non-B scope and BIOS problems are likely. The original board would've been fitted with a socket 370 Pentium 3.
Thirdly, the motherboard is powered via a series of SMD fuses on the PCI interface card. If the power draw is different on your replacement motherboard, you risk blowing those, and then you'd have two boards to repair.
Fourthly, the integrated IO aperture provides some structural rigidity to the case. There is a chance it'll become wobbly once you enlarge the opening. Tek probably had a good reason to stock half a dozen different backplate designs.
So in short, I don't think it's worth it at all.