Because the plastic is quite thin, it will not hold up well to a rotary cutter such as a hole saw or similar cutter. A trammel type cutter would likely be much worse. The only way I can think to make a cutter of that type to function properly would be to create a sandwich of some sort, quite impractical in the case of a bottle. Filling the bottle with sand might provide the necessary internal support, and not damage the cutter too badly.
Continuing along that line of thought, consider to fill the bottle with clean sand and use heat in the form of a piece of pipe or tubing, as you reference in your post. The surfaces created by doing so will not likely cause additional problems with your design as shown in the photos. A rolled edge or similar result from melting the edges may provide additional rigidity and additional material to wear longer than a bare edge otherwise created.
With sufficient heat, you would not have to have sand inside the bottle. For your project, even a little distortion of the hole will not invalidate the effect or operation of the bottle.
EDIT: Just returned from a few experiments. The first, a hole saw on a similar bottle, perhaps slightly thicker wall than a 2L bottle was illuminating. The pilot bit went in well enough:
But the hole saw caught as the cut was nearly completed and had to be torn from the bottle:
Attempting the same with heat was much worse:
All of these attempts would have fared better with internal support such as sand or similar granular matter. The heat might yet be the best, as the failure on this attempt was mostly due to lack of support from inside. The hole in the last picture that is "around the back" was a failure because I accidentally played the torch on the piece of pipe while in contact with the bottle. The flame melted more of the plastic than the pipe did, making this mistake valid only in the form of what to avoid.