S.G.
I think it is as simple as wanting the control of this. It is almost like a power struggle. Kids do potty train completely when they are ready to do so and even though he is 3 and doing well at home, he just doesn't have it yet or he would be telling you even if it isn't in time to rush him to the potty. My suggestion to your sister is take the power struggle out of it. Ask him if he has to go potty, if he says no, let it go. When he is to the point of telling you all the time at home and doesn't have those accidents 50% of the time, he will also tell her he has to go.
The only reason I can think of that he doesn't like his aunts bathroom but will go in public bathrooms is he had something in there scare him, a spider, a painting, something we wouldn't look at as scary but he does and hasn't forgotten.