I fill the ear up completely and then massage to soften the debris. And then repeat if necessary.Do you have an estimate how much liquid? I don't have anything like that pitcher, but I have a bulb syringe that probably holds 2-3 tablespoons, or 60-90 ml.
Yes, the liquid HAS to be warm. Putting cold water into the ear canal is VERY painful.
Using a syringe might (even involuntarily)put too much pressure on the ear, so I would not use it for this purpose.