As a South African, I'm surprised you don't have any visa options which fall under commonwealth agreements but I'm sure you've done your research.

The French embassy won't be able to have you married, they can only process paperwork you provide them after you got married in the UK.

My wife is American and I'm a French national, we got married here. She had to apply to a fiancé visa to enter the country but such a visa is only valid for 6 months. I'm not sure what the timescale is in England but in Scotland you can get married in fairly short notice (about 2 weeks). Once you're married you can then apply to have the EEA family visa you're referring to. If memory serves this is valid for 5 years and then you have to apply for Indefinite Leave to Remain. You also have to provide evidence that your spouse can support you. So ideally your girlfriend should have a job or evidence that suggests you won't need to apply for benefits.

You should be able to find guidance on the HMRC website. Good luck