Lesotho border, passport & visa considerations for DGT

The only possible place a passport would be requested would be at Sani Pass, very unlikely though. You wont be asked for anything at Bushmans Nek

You'll be in Lesotho for a lot of the hike, including at Sani Pass. The border is the watershed. The Lesotho border post is right at the top of the pass, so yes you could just walk right past it, maybe 100m or so inland
It would be safer to get a visa though (I'm assuming you are not South African?) as technically you'd be illegally entering Lesotho

Welcome to this forum caiokf. I see from your profile you are not South African, so I thought I would emphasise some details

As others have said, rarely do people on a Grand Traverse need their passports. However you normally walk right by the Lesotho border post at the top of Sani Pass, and the South African border post at Bushman's Neck. There isn't really an official entry and exit point along the route. It would only be an issue if you wanted to end your hike at Sani Pass.

Nobody can guarantee that you will not need your passport with a valid visa though, so do consider what others have already warned you of in this thread. If you stick strictly to the escarpment edge you are very broadly on the border, so if the Lesotho officials approached you, they might simply just tell you to get back into South Africa. You cannot take this for granted though. Also bear in mind that the border is not well defined or understood either. There is a dispute sometimes between the watershed and between the actual escarpment edge, which isn't always the same thing.

If you decide not to get visa, definitely still take your passport along. If you are questioned by the Lesotho official’s (they do patrol on rare occasions) just be as co-operative and respectful as possible.