Does selling an item at a lower price in the context of another sale fall under the prohibition of a Muslim selling over his brother's sale, and are there any regulations for it?

Merely offering a product at a lower price does not fall under the prohibited act of "a brother selling over his brother's sale," because an offer is not a sale. Forums where products are displayed for sale are subject to the same rulings as a marketplace. Merchants display their goods at varying prices, and there is no religious impediment to this; it is considered commercial competition.