Can that camera be used in APS-C mode with a 70-200mm lens?
Sure, it can. But it probably won't give you much of any advantage over shooting in FF mode and cropping later. The only improvement will be the number of images the camera's memory buffer will hold before the frame rate bogs down waiting for files to be written to the memory card. You won't get any faster initial frame rate before the memory buffer fills up.
To get more detail of a small subject you need one of two things:
- Higher pixel density with the same focal length
- Longer focal length with the same pixel density
Shooting in APS-C mode gives you neither of these things. It only gives you fewer pixels used to show a smaller area surrounding your relatively small subject compared to using the full frame image to show a wider field of view.
Cropped to APS-C, whether in camera or after the fact, your Sony α7 III gives you 10.67 MP images. To get 24 MP in 1.5X APS-C crop mode, you'd need a full frame sensor with 54 MP.
If you were instead to use a 300mm lens with the Sony α7 III, you'd get the same field of view as cropping to APS-C with a 200mm lens, but you'd have 24 MP instead of 10.67 MP, which means you'd be able to see finer detail in your subject assuming the 300mm lens can resolve as well as the 200mm lens can.