I've been thinking about this a bit.
It happens at the exact same time as
@Fantastische Hure alludes to in his original post.
News travels slowly across space and Cooler only just finds out about his brother dying on Earth. He's shown a video of who did it (Trunks, not Goku) but believes he
is Goku.
He travels to Earth along with his squadron. They sense the arrival and gauge the level of the fighters to be around Perfect Cell (or maybe even more powerful, not accounting for Cooler who they can't sense yet).
Everything is tense and eventually the Armoured Squadron arrives where Gohan, Piccolo and Krillin are. Unlike in the movie Piccolo is bodied in one hit by Doore, Krillin is taken down second.
Gohan transforms into a Super Saiyan and holds Doore off for a little bit. That's when Vegeta and Trunks both arrive from Capsule Corp and defeat Doore and Neiz in a three on two scenario.
Salza arrives and he's able to hold them off for a bit but before he's ultimately defeated he runs off to get Cooler.
Cooler arrives in his first canonical form. He one shots Salza for being weak and failing him.
He bodies Vegeta, Trunks and defeats Gohan with slight difficulty. That's until Gohan turns SSJ2.
Cooler transforms into his final form but because of the rage boost (Cooler here is probably Super Perfect Cell level) Gohan bodies him.
Gohan ends up having a beam clash like he did with Cell overpowering Cooler.
The end.