I agree with Kevin, small private camping trips are not possible. Nor desirable, I should add. The environmental impact of tourism increases significantly with people camping - such as waste disposal, especially human waste, as well as giving the residents (wildlife) some peace.
IAATO ( the International Association of Antarctic Tourism Operators) does a generally good job at self-managing tourism impacts (e.g. use of drones).
I think that small ships, like that used in the Antarctica 2020 Expedition, are the best of both worlds. They provide a personal convivial environment, as well as good food, a comfortable bed, and reliable waste disposal. If you want a camping experience, I suggest there are better (and less sensitive) places to do this.
One more thing, most tourism operators aim to maximise the sights, often at "convenient times". This means that photography can be hurried and often not in the morning/evening light. A true photographic expedition is more focused on morning light than a comfortable sleep-in! So it looks like the A2020 Expedition is the ideal mix!
Antarctica changes you. Shackleton's navigator spoke of the "little voices" bringing you back.
Its likely that you will become an Antarctic Ambassador. This Continent needs Ambassadors to speak on its behalf when those making decision remotely have never experienced its grandeur!