Is it acceptable for us to apply for a UK visa for my daughter on her South African passport, as I don't think her British passport will be processed in time for our departure?
You can apply, but you might as well donate the visa fee to the home office and save yourself the trouble of preparing the application form and supporting documents.
The UK has a visa-like thing called a "certificate of entitlement to the right of abode," which it issues to people who have the right of abode but do not have a UK passport, including dual citizens.
The guidance notes
If a person who is a British citizen, or has the right of abode in the UK, applies for a visitor visa or for settled status to be entered into a foreign passport, the application should be refused. ... If you have a visa application from a person who you are satisfied has the right of abode in the UK, the person should be advised to apply for a certificate of entitlement instead.
Now you might be able to get a visa for your daughter pursuant to the next paragraph, which reads
In some cases you may have a visa application from a person who you believe may have the right of abode in the United Kingdom, but cannot establish their claim. If you are satisfied that they have made genuine and reasonable attempts to try and obtain the relevant documents but cannot do so, you may issue a visa in line with the Immigration Rules.
I don't think this applies to your daughter, because her claim to British citizenship seems to be clear (assuming that you were born in the UK). Still, I could be wrong about that: the clause about "genuine and reasonable attempts" might in practice apply to people who have a clear claim.
You might want to look into whether your daughter could receive a certificate of entitlement without having to submit her mother's unabridged birth certificate. If so, she could use that to travel to the UK. The guidance (pdf) suggests that it may be possible, since it requests only the "parent's full birth certificate, registration or naturalization certificate" (at the top of page 11). That might indicate that they only need the relevant document from the British citizen parent, but it might also be a misplaced apostrophe.
The certificate of entitlement is rather more expensive than a passport however, costing £388 when the applicant is not in the UK.
