The first thing you don't want to do in that situation is go IMC till the emergency is absolutely under control. If the fire doesn't go out, I don't know what your orders say, but mine say ditch. Single engine IMC with a fire is not where I'd want to start from to get on the water.
Whether to go for the platform straight off or shut down first would, I think, be one of those "It depends" calls. However, a mile out is probably over a minute to wheels on deck - thats a long time with an engine bay burning merrily. I think I'd want to be a lot closer than that - like very short finals - before continuing twin-engine.
Sven