Why does my LM7805CV output voltage drop at loads <= 150 mA?
Measuring with an LED and 330 Ω resistor is okay. So if you did that measurement correctly and it only provides 4V then it doesn't meet the spec in the data sheet you linked. (Whether that's the spec for that actual part is unknown since the manufacturer is unknown.)

When measuring with that variable load device, you may need capacitors depending on what exactly the load is doing.