From an electrical point of view it would be best to have a specific 16A radial circuit for the heater on a higher rated (16A+) plug and socket arrangement.
The rating of the immersion means that it is pulling the maximum current that a BS1363 socket should take so any slight anomalies may cause melting such as you have seen. The same thing happens with standard immersion heaters on plug and socket or fused spur arrangements.
With the circuit installation you have you are unable to avoid requiring a 13A fuse at some point, presumably the wifi socket may be fused, the plug in the wifi socket will be fused and in the current set up the plug on the outside is also fused.
Each fuse (and each connection) is a potential point of failure.
The wifi socket must also be close to its rated limit as they are often limited to 10A or even 5 A.
The simplest solution would be to have a 16A socket installed outside instead of the double socket and still plugged in as you have it now inside, however this will only transfer the possibility of failure to the wifi socket and the plug which is used inside and will not address the heavy loading on the ring, although if it is only used at night the loading on the ring will probably be low.