So thats a ping of 240, minimum. That's assuming everything else in the system happened instantaneously.
If you can however manage to solve the whole aiming at a moving target 360 kilometers away then maybe you could do a LEO network, and yeah the ping would be closer to 1.2 assuming again literally everything was instantaneous.
As you note, LEO is very different from geostationary. Moving target doesn't strike me as impossible. A LEO satellite has a very predictable path, a stationary dish has a predictable movement relative to that path and a mobile phone knows where it is at all times.
There are several companies building (and a few shipping) flat-panel phased array antennae that can "point" at a moving target 360 km away. They also can switch to a new satellite instantly.
36000 km / speed of light = .12 seconds
So thats a ping of 240, minimum. That's assuming everything else in the system happened instantaneously.
If you can however manage to solve the whole aiming at a moving target 360 kilometers away then maybe you could do a LEO network, and yeah the ping would be closer to 1.2 assuming again literally everything was instantaneous.