Did you ever watch Red Green do an oil change on a car?
I saw him do it 2 different ways: roll the car over and dump the oil out the fill cap, or pry out the engine and turn the engine over to dump the oil out the fill cap.
So....there you go. 2 options. You can replace that screen -- while keeping your oil in the engine. And sleep nights.
R.G. would be proud of you!
Stig