How to add a new movement "mode" to Lyra?

Yes you can, there are many ways to do things. For a sprint, I would do it the way you have it there. However following how they are doing it with their weapons adding a new layer makes it super simple. I duplicated their rifle layer and tweaked it to my weapon. The weapons data has a variable for the layer file, when weapon is equipped the layer is used… very clean.
For anything standard locomotion I agree doing it the way you example is better.