I managed to fix the issue by generating a new Collada file.
In the original version exported from Blender, I used a constraint to produce the animation, and Blender's 'bake action' feature to 'freeze' the animation. I also chose Blender's standard Collada exporter to produce the Godot scene.
For the fix, I used the same Blend file, but used the official Godot Collada exporter,
which appears to bake animation constraints on export automatically.
I still need to perform more tests, but it's certainly running faster than it did before.
Perhaps Blender's standard exporter was generating too much redundant data, slowing down the renderer.