As a dual national, can my daughter travel to the UK on just her British passport?

As far as I can find on the net, neither South Africa nor the UK have any problems with dual nationality acquired at birth, so you don't need to be particularly hush-hush about it.

Bring both sets of passports and present

• To border guards of one of the two countries: The passport of that country.
• To everyone else, including airline agents: The passport of the country you're traveling towards.

and have the other one ready to show too if someone questions whether the one you show will support your entire itinerary.

Remember that immigration rules generally apply to whole persons, not to a particular passports of theirs. As a citizen of both countries, your daughter has a (near) absolute right to travel back and forth between them, as long as she has valid documents to show those citizenships.

Rules that a passport must be valid for so-and-so long after a trip generally apply when you're entering a different country than the one you're a citizen of. Basically they want to be sure that if they need to send you back, they can do so without complaints from your home country that the passport isn't valid anymore. However when you're a citizen of the country you're entering that is not a concern -- they will not be sending you back involuntarily anyway.

If she's entering South Africa on a South African passport, the fact there are only 3 days of validity left shouldn't matter, as she is a national and therefore has a right to enter the country. If the only issue is how to get onto the flight, and they don't accept a valid SA passport, you could always board the flight using the British passport (which you don't need a visa for), and then present the SA passport at SA immigration - there's no need to show the same passport at immigration that you used to board the flight. Both the UK and South Africa allow dual nationality (with the proviso that you must use your SA passport to enter South Africa), so there is no problem with having both passports on you.

She should only show the South African passport to South African officials - if asked how she will enter the UK without a visa or anything then present the British passport as well, however make sure that he South African one is presented to avoid any possible accusation of attempting to enter or leave the country on a non-SA passport.