I knew that London wasn't cheap but I've no words!
40-50€ is just one price here in Athens. It seems that the engineers have a scanner, and if they smell money in your pockets they're adjusting their prices accordingly!
Anyway, apart from labor cost, parts & repair time will be really useful for the rest of us, to do our math! (and probably any DYI plans).
A corroded or even a green verdigris crusted connector can cause the heating of the terminal and the fuse blade due to high resistance. The starter relay is doing nothing once you release the starter button except passing power out from the battery lead to the smaller loom connectors via a hard wired connection and allowing regulator input back to the battery. Any non insulating grease such as coppaslip or a dielectric silicone smeared into the whole area where the connector and fuse fits should prevent this happening.
The damage to the red connector looks more like it's external from the