How to book car hire online for South Africa with permission to drive through Lesotho / Swaziland included?

I gather that different South Africa car rental companies have different policies on taking vehicles out of the country, and that most allow it for Lesotho and Swaziland but need to give a letter granting permission to take the car across the border, which costs extra.

That's fine, but I can't find any car rental website for South Africa that mentions this in their options, terms or FAQs or that gives any facility to add this to an online booking. I've found accounts from people who've done it (example on TripAdvisor forums) but they seem to all be people who just turned up and hired a car at the desk at the airport. I'd like to have something booked in advance, and pick it up somewhere other than the airport.

How do you obtain permission to take a car through Lesotho from a South African car rental company when booking online? And how do I know in advance while choosing between car rental companies how much it'll cost?

Here's what little I've managed to find:

• Avis South Africa have an incredibly vague note that this "could create some complications. At Avis, cross border travel options can be made available upon request", then a vague mention of "local agents".

• First Car Rental have reasonably detailed info in their FAQs, even including the fee for Lesotho (R750) and other countries. They say "A letter of authorisation must be requested at the time of reservation for all cross-border travelling", but don't give any more info on how to request such a letter. It also mentions needing "An affidavit from the police" but it's not clear if this is something they provide or something I'd need to get from the police separately.

• I haven't found any car rental comparison site that has anything about cross border travel policies. Closest so far is Rentalcars South Africa, which acknowledges that cross-border travel is a thing and that company policies vary, but cops out and suggests flying then renting a car in each country... their terms and conditions contains this which seems to be generic but isn't entirely clear:

  Cross-Border fee - ZAR 1,320.00 including tax... Customers must inform the car hire company of any cross-border travel

Update: I found some more detailed info in AVIS's terms and conditions under "Cross Border Rentals; this document seems to become available only when you start the booking process, possibly based on your country of origin (that, or I did a very bad job of finding them on the first attempt). They don't say anything about how to get it, but they do quote:

Additional One Way Cross Border Fee... ZAR 1330.00 (incl tax) into Lesotho

Based on it, I'm trying just doing the booking online then sending a follow-up email requesting cross-border permission. I'll update this with what happens... Wish me luck...