I have used the exact setup you are using. I found that the 2X Teleconverter iii worked wonderfully with the 70-200 f/2.8 IS II on the 5D Mark iii. I actually tested it in far harder conditions (shooting a wedding in a relatively dimly lit gymnasium) and it worked quite well. Particularly since you will be using the shorter focus distance setting, I wouldn't expect too much problem.

The focus was slower, but not significantly enough that I'd worry about it, even for your use case. Image sharpness was still pristine. It was certainly lower than without the converter, but more than usable. It was far higher quality than my 70-300 f/4-5.6 IS that I was using prior to getting my 70-200 f/2.8 IS II. The consensus here is quite correct, there is a little (but liveable) corner loss, but the center is very clear still.

That said, if you are willing to spend more, the 100-400 f/4L will beat the 70-200 f/2.8 IS II with 2x teleconverter in terms of quality and speed, but it also costs 3 times as much.

For $450 the teleconverter is a great thing to have in your bag of tricks to go with your 70-200 f/2.8 IS II, particularly for the type of shooting you are doing. In fact, the 70-200 f/2.8 II is one of the few zoom lenses I'd recommend the 2x III for because it is so clear. I have to admit I have not purchased one yet, but only because I don't shoot at ranges where I need it often enough and have more pressing upgrades for my kit. Both the 1.4x III and the 2.0x III are on my list of things to get eventually.