Have you tried a POST wake_up before the POST charge_stop?
It shouldn't need it, as it seems that the car is always awake when charging, but you never know!
It would be nice if Tesla would just endorse the use of their API officially, so we had solid info on how it works, especially as it seems that there are now lots of app using the unofficial API anyway.
If I had to guess, I'd say the car is fine with just a wifi connection, as it seems to prefer to connect via wifi if it can (just going by it always connecting to my home network as soon as it's within range).
Like you, my guess is that all the GET stuff comes mainly from the recorded status held on the server, and doesn't bother to make any calls to the car, unless, perhaps, it's already awake.
BTW, I'm doing something vaguely similar, but using micropython on an ESP8266, just to make it a bit more challenging...