Let's take this scene as example:

Node Tree
By default, when you enable using nodes, Blender will create a Render Layer and Composite node. It is important to delete the Render Layer node: otherwise, Blender will render it.
No Render Layer node = no 3D render.
All you need is an input node (movie clip in my case, it can be an image sequence too, it doesn't matter). Then the Kuwahara node, then the Composite node and Viewer.
Viewer is just for previewing and doesn't affect the render.
Composite sends the composition to the render. When the render is done, Blender will always write a file as per your scene's file output settings. This is why you shouldn't replace it with a File Output node: Blender will output two files instead of one.
At least, it will until the next Blender update or so:
#117129 - Render: Skip writing output if no Composite node exists - Blender Projects
Compositing Options
In the Sidebar, Options tab, you may want to do a few changes:

By default, the chunk size is set to 256. This setting splits your compositing area into smaller chunks, that are usually faster to render, but at the cost of more overhead time. You can do tests to see which chunk size is faster for you, usually the smaller, the better. 32 worked the best for me.
Enabling OpenCL might give faster performances depending on your hardware.
If you had a more complex node setup, you could also enable Buffer Groups, which would allow you to put a few nodes in a node group and have it basically act as a cache. But it's useless with just one node to process.
Output Settings
As mentioned here:
What image format encodes the fastest, or at least faster? PNG is too slow - Blender Stack Exchange
Different file formats, and compression settings, give very different writing speeds. PNG is of the slowest, especially with maximum compression.
The choice of format also depends on what you want to do after. If it's final delivery as a video, you could honestly just output a video directly (you can add the audio in the VSE).
If you need editing after, you could export as TGA, or even TGA Raw / EXR if you want to do some color management or stuff that need high bit depth - and if your input data is also already high bit depth.
I noticed that the render percentage at the bottom goes to 100% twice before outputting an image. I assume the first is just rendering my viewport and the second is my compositor. So should I reduce the render settings on my viewport to super low to avoid longer render times for my compositor?
– Kris Theorin Feb 09 '24 at 17:42