My recommendation is to start with the business process to manage servers, automate, and feed that data to TBM.
I did have to maintain a manual mapping a few years ago. Back then we coordinated a naming convention for servers that allowed us to map servers to apps, but manual really does not work (at a reasonable cost).
This is a task for the CMDB team. They are the experts and should take responsibility for it. This is a task much larger than TBM, with great benefits to TBM but not driven 100% by it. There are automated network sweepers that do many more things than just map servers to apps and resolve the issue in a heart beat. We use ServiceNow.
More important than the tool, or at least as important, is the business process, and discipline to follow it, to manage servers in a way that the tool of your choice is effective.