The physical size of the heater doesn't dictate its power rating, if your solar system is rated at 600W then your heater will need a power rating of less than this.
Do you have the power rating of the treadmill and exercise bike?
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Assuming the output of the inverter is properly protected for overload and short circuit then there would be no point installing addition fuse or mcb protection.
An RCD will only work if one pole of the output of the inverter is connected to earth permanently and reliably. In my opinion for such a small inverter it would be safe enough, if the installation is designed suitably, not to do this and keep it as an electrically seperate system.
I can find out the power rating for bike later this eve. Just sold the treadmill as it was s***. I may purchase another but for time being I'm going to stick to road running.
From speaking to the company where I plan to purchase the Solar kit from, they have assured me its a very reliable & internally protected inverter so I'm confident it should be fine to run the lights/ipod dock safely.