http://www.wotmeworry.org.uk/manuals/Vespa/LX150/LX150Workshop-searchable.pdf
Electrical info starts at page 57 I think you want to look at page 61 & 62. It shows 2 15amp fuses coming off the battery, 62 shows the starter relay. When you apply the brakes 12volts is sent to the starter switch, press the starter button and the starter relay is energized. It closes and connects the starter directly to the battery, no fuse inline. So if the brakelight works you need to look at the starter button, the starter relay, and last resort the starter itself. I would use a voltmeter and check the starter relay. See if you have power on terminal 87 when the key is on, the brakes are applied, and the starter button is pushed. If no voltage is measured, go to terminal 86 and repete. If you have 12 volts the relay is bad, if not the starter switch is bad. The other diagrams show other 15amp fuses that I think are the same ones in different diagrams. Hard to figure out just how many there are.
Scooter troubleshooting starts at page 49. Its worthless!
I had Staples print my manuals, double sided and spiral bound. Makes it easy to work on the scooter without dragging the laptop outside and running the risk of damaging it. It can be done for less than 1/2 hour of tech time at the dealer. Good luck