> you're not going to make a separate online order with a fixed-overhead delivery fee from the less-upmarket store
If I were doing this, I would be far more likely to make a separate order online than physically. It takes no effort to do it online, but to shop at more than one physical store is additional inconvenience.
If you've got a Whole Foods and a less-upmarket grocery store equally far from you, you're gonna get your staple foods from the less-upmarket store, and just go to the Whole Foods for the things only they carry.
But if you're making an online order with a fixed-overhead delivery fee from Whole Foods, you're not going to make a separate online order with a fixed-overhead delivery fee from the less-upmarket store; so you're going to end up ordering the staple foods from Whole Foods, too.