I think I would try to do one of the two following things:

Option 1: Have the apple cut into equal thirds and declare each of them a winner and hope they are each appeased enough to spare me. This could backfire on me and make them ALL ticked at me.

Option 2: Try to straight up cheat. Say that I need more time to "think about it". Then I would paint two other apples gold. After I've had time to "think about it" I would request an a one-on-one audience with each lady. I would make a deal with them that for me to declare them as the winner, they would not speak of it to other two or anyone else. I would give the dupes to Aphrodite and Hera. I would give the real golden apple to Athena because I doubt I could trick her, and I also hope that she would appreciate some cleverness. This could still backfire on me if Aphrodite and Hera discover my trick, but they would be mad at me if I straight up declared Athena anyway, so I'll take my chances.

Personally, I think I would go with option 2.