score:2
Just got back and it all seemed to work fine.
To get the permission letter, basically you just:
I emailed in advance to request permission for Lesotho, and then decided while at the desk that I wanted Swaziland too. The only difference was, it was the guy at the desk who said "And you still want a letter allowing you to go into Lesotho?", and it was me who said "And I also want a letter allowing me to go into Swaziland".
They got the message from my email, but having told them in advance didn't make any real difference.
Using the permission letter is pretty simple:
When crossing into Lesotho one of the officials on the South African side, while checking I'd stamped my passport, said "That's an AVIS car, isn't it? Do you have the letter?" then just glanced at it. It's clearly a pretty common situation.
When crossing into Swaziland, they didn't even ask to see it. I'd certainly recommend having one anyway, don't want to risk being mistaken for a car thief... It's also possible that they didn't ask because I was visibly holding a big bundle of AVIS paperwork along with my passport etc.
Random tip: if you're crossing into Lesotho at the border near Maseru and it's busy, don't do what I did which was drive past all the full parking bays looking for an empty one to stop and get my exit stamp, then end up stuck unable to reverse or turn around in the tight space between end of the parking bays and the border itself, with an impatient horn-blaring truck stuck right behind you and about 20 South African police and border officials looking on shaking their heads... (though this clearly happens quite a lot: a laughing official let me use the staff car park after wryly saying "This is your first time here, right?").