What BNC jacks are feeding the monitor?
R,G,B, H, and V or R,G,B, and composite H-V sync?

If it has separate H and V sync, you can buy a BNC adapter cable and
use any computer monitor that is capable of displaying the 1280x1024
resolution at the same vertical scan rates.

If it uses composite sync, you need a different adapter cable that is
more expensive and includes a sync separator.

It is not practical to try and use a laptop display for this as they
need a digital signal from the laptop motherboard.

If that is really that critical of a system that it cannot be down,
they really need to invest in buying a spare monitor for the system.
Even a good used working one will suffice, many different companies
have similar fixed frequency monitors available with bnc inputs.