Hi - observation is the 13A plug melted when used at about 13A. A 13A fuse won't blow when there's a 3kW load and no fault, but maybe there was a bit of corrosion/tarnish on the 13A fuse and that made it a bit more resistive and so it ran hot? Anyway, perhaps a new circuit from your CU with 16A socket and plug would fix this?
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Would a 16A socket and plug suffice? - running a new circuit off the CU would be a ball ache due to its location.