How good is the idea to place mostly entire copper layer up and down the functional layers in PCB?
The PWM frequency might be 5.6MHz but that doesn't mean it will actually turn ON and OFF LEDs at 5.6 MHz. Also, PWM having a PWM frequency that high in your application is also very unlikely.
Basically your PWM will get converted to average DC value and that will drive your LEDs.
Coming to your question about placing extra layers on top and bottom of PCB, they will definitely help you reducing the radiated emissions if your application requires so. But if there is no concern of RE, I don't think it is necessary.
Also, if your current requirement is not being fulfilled by your current traces that are driving LEDs or are on the edge of the specs, then you should add either extra layer or you can increase the width of the traces if the space allows on the existing layers.
If you place those extra layers, you can connect it to Ground to make best shielding out of it or you can also use it as your redundant supply path.