At the moment Fuse 2 is supplying the contact and also the coil via the neutral switch
This is a weak supply, too much resistance hence the ignition switch melted if not that it will certainly be prone to Startus Interruptus
The relay coil only draws about 1/4 Amp not a problem for the ignition switch.
The solenoids have 2 coils, the Guzzi factory don't seem to know that because they never show the second coil on their schematics
The one they show draws 10 Amps, the one they don't show will draw 40 - 45 if its wired properly, at best wired through the switch you might get 25 and it's working half heartedly, but its fried your switch.
This high current only lasts until the solenoid has pulled the gear into place and closed the contact.
To answer your question directly
The new fuse feeds the relay contact, fuse 2 feeds just the relay coil via the neutral switch.
This is something Guzzi have done every now and then on some models for example 2000 era spine frames and early Grisos
I took the time to open up a Bosch solenoid to show the two coils, The Valeo is similar.
This is a Valeo on a 2001 VII Sport but the Bosch solenoid is almost the same
Take no notice of how the relay coil is wired keep that as per the T3 diagram.