Modern Vespa

Hell Everyone.
I write for help to solve a problem I'm having with the electric starter not working on my Vespa LX 50.
If I use the electric start when the engine is cold, it turns and the engine fires up OK.
But after, when I use the scooter for a long enough ride that makes the engine get hot. If I turn off the engine and immediately try to start it again, the electric starter doesn't seem to be sending any power to the starter mechanism at all. When pressing the switch I can only hear a very quiet electric relay click coming from under the seat area. I can't hear any sound at all from the starting mechanism of the engine.
However. if I leave it for a good while, say an hour, after the engine has cooled down, when I try the start again, it turns the motor and fires up OK.
I have the rear brake light coming on as it should to make the contact for the starter and the battery is brand new (replaced 4 days ago as I previously thought that was the problem).
I have also put in a new bendix starter motor.
I don't have a multimeter to provide you with some voltage readings at the moment. But in the meantime, until I have bought one, could anybody advise me on the solution from the symptoms I provide here?
I already seem to have unnecessarily bought a new battery and bendix starter motor when they weren't the actual problem. So now I'm focusing on that the problem may be from the voltage regulator or the stator plate. As both of these items are not cheap, I would really appreciate any help to narrow it down to either one of these, before I change the wrong thing again.
Thanks a million for any help

It may be your starter solenoid/relay. When it gets hot it stops working, that could explain the clicking like its a low battery. You should test it and see.

Here is a thread that discusses it: Piaggio BV 500 starter relay question

Here is another one: 1998 ET4 Clicks once instead of starting

Let us know how it works out.

HI Pinkscooter.
Thanks for the help.
i'll have a look at that and let you the results.
Best regards.
Max