I;m assuming you have one of the later VIIs with the Start relay powered from the Ignition Switch
Typically there is too much resistance in the ignition switch and it's associated wires. The Starter solenoid has two coils one that pulls 10 Amps and another that would like to draw 40 Amps but just for a few milliseconds while the solenoid engages and the main contact closes, This is such a common problem it's been named "Startus Interruptus", common to almost every other Guzzi right up to the latest.
You need to clean the ignition switch every few years, take the switch block off the lock (2 Phillips screws) and open it up, wipe out the old grease and replace it with fresh Petroleum Jelly aka Vaseline.
There is a permanent fix (strong feed) that requires a direct connection to the start relay 30 terminal from the battery but it also requires a different arrangement for the feed to the headlight relay, try cleaning the switch first. Drop the negative wire off the battery to make it safe.
I think your problem with cutting out is possibly a dirty side stand switch, carry a short length of wire to stuff in the relay (15) socket 30-87, this will bypass the stand switch as a test.
Further reading
https://www.v11lemans.com/forums/index.php?/topic/21206-startus-interuptus-revisited/