This sounds a lot like shader compiling (See Shader.CreateGPUProgram takes a long time on first startup (iOS) ).
When you take a look at the profiler to see what exactly causes the spike, do you see “Shader.CreateGPUProgram”?
A screenshot of the attached profiler to your iOS device with a complete hierarchy foldout of the profiler frame which shows the spike would be helpful to understand the cause better.