"The physical config was accidentally overwritten and there are no backups".
Welcome to legacy.
So for a situation like this, there are several things that you need to think about. First and foremost... what is the impact when (not if) this process finally stops? This isn't just for technical people. You need a business impact assessment. You need the users involved. They're your lever for fighting the inevitable fear-based political hurdles. Is it an annoyance? Or does the company go out of business? The potential severity of the impact matters a great deal. If it's putting the entire business at risk, you should be able to get support from the highest levels of management to do whatever is necessary.
Second... how do you recover? There are a variety of ways off the top of my head. The most obvious would be to reconstruct the physical config. The "obvious to others" that is probably a stupid idea is rewriting the application. Let's ignore the stupid one and start dealing with reconstructing the configuration.
Do you have the source code for the system? If so, you can probably reconstruct the configuration architecture from reading source, at least. It may suck, but it's something.
Is there a test environment with its own running copy of the app? If so, it will have its own configuration, which will make reconstruction much easier, as then you differ only by values and don't have to figure out what the fields are.
Now, what kind of data is in the configuration that makes it difficult? Resource locations? Authentication credentials? Something else? If it's connecting to external systems, you can look at logs, packet-sniff, etc, to at least figure out where it's going. Credentials can be reset for a new version - a painful one-way trip, but it can work. Do you own any external systems, or are they outside your control?
Now, all systems have inputs and outputs. What is the output of this? Is it going to a database? If so, are you backing up that data? Make sure any locally stored data is getting backed up!
If there isn't a duplicate test system, what would need to be done to create one? Are there licensing restrictions? Specialized hardware/OS? Are you building from source code? Do you have the source? Do whatever it takes to create a parallel system that you can test configuration on, make it run elsewhere.
I can just go on and on with this, but the important thing is to be able to duplicate as much as possible before you try to replace. And find out what the cost is - that buys you authority.