I don´t know specifically about a mazak, but most-alll real backup programs can make binary images that are absolutely identical to the original hdd contents.
They can then clone hard drives onto same-size or larger new hard drives.
It is possible the mazak might check the hard drive manufacturer and or serial number and note that the drive has been changed.
Microsoft does this, for example on windows 7++ and office programs.
No idea how likely this is with mazak.
Aomei backup standard is one such easy to use program, cheap.
Install aomei on a pc, plug the mazak harddrive into a spare pc hdd connector, make a binary image onto your pc, then make a second identical hard drive image onto a new hdd for the mazak.
I would suggest an identical sized/type hard drive.
Then try the new hard drive in the mazak.
Should take about 1 hr first time end-end and under 30 mins once you are used to it.
Disc-disc transfers are really really fast- about 12 mins for 200 GB on SATA solid state drives.
Keep the hdd image on your PC.
It may (will usually) be possible to mount it and read individual files, unless it uses some really weird file system.