How can I get into Lesotho when I'm banned from South Africa?

According to South Africa Consular: Civic and Immigration Services (North Americas), it does allow transit without a visa. However, there is an exception, the last paragraph in bold, which could affect you.

Travelers transiting through the following international airports are exempted from/do not have to be in possession of transit visas, but will be subjected to biometric capturing directly at the airport:

1. OR Tambo International;
2. Cape Town International;
3. King Shaka; and
4. Lanseria (Johannesburg)
   

Please note that the above-mentioned exemption does not apply to deportees. Please also note that foreigners who are utilizing land ports of entry to transit through the Republic of South Africa and who are subject to South Africa visa control should apply for and be issued with port of entry visas.

Although you mentioned a ban, not deportation, contacting a Consulate would be advisable. As of yet, I've been unable to identify an air route that does not transit through a South African airport.

All depends if you need to clear through the immigration as not sure if it's consider internal flight. According to UK FCO if it's just a transit you don't need and in the case I described they suggest you check with the airline. https://www.gov.uk/foreign-travel-advice/south-africa/entry-requirements