Actually, I have experience here.
The problem is that anything you do that's potentially destructive in service of getting the system to be more sustainable is going to be met with heavy criticism. So you must be careful, the company has accepted the risk that they're in and you'll have to contend with that most likely.
First thing is first: is it a VM or a Physical machine? Things get a little easier if it's a VM because there might be vMotion or some kind of live-migration in place meaning hardware failure might not actually bring the service down.
Next thing is you absolutely have to plan for failure because the one thing that I learn as I learn more about computers is that they're basically working "by accident", so much error correction goes into systems that it absolutely boggles my mind and failures still get through. So, it's certainly not a question of "if" but "when" and plan for it being soon.
Now, the obvious technical things are:
* Dump the memory
* Grab any files it has open from its file-descriptors (/proc//fd/) Your config file might be there... but somehow I doubt it.
Attach a debugger and dump as much state as possible
Be sure to cleanly detatch : https://ftp.gnu.org/old-gnu/Manuals/gdb/html_node/gdb_22.htm...
Don't use breakpoints! They will obviously halt execution.
If it was my service I would also capture packet's and see what kind of traffic it receives most commonly; and I would make some kind of stub component to replace it before working on the machine. Just in case I break it and it causes everything to go down that depends on it.
But, this is a horrible situation, reverse engineering this thing is going to be a pain. Good Luck.