You shouldn't be looking at project management, except possibly for inspiration. What you are really talking about is process engineering.
Projects vs. Process
Projects are generally understood to have a defined duration and a set of exit criteria. Processes, on the other hand, can be ongoing. A project contains processes, and project management is typically understood to be the art of applying controls to those processes. A process, on the other hand, can exist independently of a managed project.
In your case, you might find some of the process controls from Kanban or Lean Manufacturing to be useful in designing your processes. However, you don't necessarily need the whole framework if all you want to do is apply some of the controls and principles to your own processes.
Designing or modifying a process is the domain of process engineering. Many of the agile principles are useful in process engineering. For example:
At regular intervals, the team reflects on how to become more effective, then tunes and adjusts its behavior accordingly.
This is the core tenet that is often restated as the "inspect-and-adapt" cycle, and I certainly recommend adopting it to ensure that you periodically review the operational effectiveness of your controls.
There are also other principles lifted from Kanban, Lean, and Scrum that may help you. These principles include:
- Reducing different kinds of waste (muda, muri, and mura).
- Optimizing for throughput instead of utilization.
- Reducing cycle times.
- Making procedures measurable and repeatable.
- Ensuring processes and procedures are testable.
Whole books have been written on process engineering, so this can only scratch the surface. As long as you inspect-and-adapt, though, you can't help but benefit from the empirical knowledge you gain from any process or control you apply.