If you are having a three pin power connector (socket end), then the large or middle connector is ground. But you need to ensure that it is physically connected to the earth. To check, connect an electrician lamp between the power line and the ground and you will see the lamp glow at full brightness. Testing with a neon tester is unreliable. In most countries, it is not legal to connect the neutral to the ground wire at the consumer end. They are usually connected close to the distribution transformer but the ground is always locally connected.
I too do not understand why the power strip ground is not usable. If the other end is connected to a reliable earth point, the power strip ground is a good ground.
Conducting rubber pads can be connected to the ground directly or via a high value resistor. A line conditioner removes high voltage short transients and is always welcome.
Most likely the green LED shows that the power is present (it is connected between the line and neutral) and does not indicate the status of the ground pin.
My personal suggestion: test test and test. Do not assume anything when it comes to your safety or the safety of your work!