When you turn the headlights on you are applying more load to the alternator so the belt works harder, so the belt slips.
Just take the alternator to an auto electrician and get them to change the pulleys. It isn't hard. They do this sort of stuff all the time.

Oh okay - So once the pulley is changed the belt should be able to handle the extra load a little better?

Alright I'll give an auto electrician a go first. I've emailed convertible mini, see what he says.

Cheers, I'll keep you guys posted

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I wish I had some sort of witty quote in my signature

Yes. It will work. There is nothing wrong apart from the wrong pulley.
Even your old belt will work.

Well fingers crossed then! Just one problem left, and that's the fan belt slowly loosening itself off every now and then.

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I wish I had some sort of witty quote in my signature

The other option is take pulley to machine shop or someone with a lathe and a parting tool and ask them to machine the bottom down deeper so the Belt does not ride on the bottom and slip. It does not matter if the belt sits a bit low in the V as long as it only rides on the sides. This only depends in how much meat you have and how low can you cut it. Turn Belt inside out and sit in V to see how much clearance you need in the bottom. Belt should not touch the bottom flat of V. Or post me the Pullet and I will see what I can match it to.

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DID I MENTION THAT I AM THE BEST SUPPLIER FOR MINI ELECTRICAL PARTS AND REPAIRS?
"Let us remove your "SHORTS"

Many auto electricians have/had a bin of pulleys stripped off old alternators. I got one there once to replace my worn one.
Avoid the pressed steel ones, the cast iron pulleys wear much less.

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DrMini- 1970 wasaMatic 1360, Mk1S crank, 86.6HP (ATW) =~125 @ crank, 45 Dellorto (38 chokes), RE282 sprint cam, 1.5 rockers, 11.0:1 C/R.