Is it permissible for me to ask a customer to search for other offers, and then amend my offer based on that, and is this considered among the impermissible “buying over buying” [bayʿ ʿalā al-bayʿ] in Sharia? And what are the cases in which “buying over buying” is permissible?

There is no harm in you saying to a customer or worker: "Look at the market price, and I will sell it to you for less," or "I will rent it to you for more." This does not fall under the prohibition of "selling over the sale of one's brother" or "bidding over his bid." What is prohibited is when two parties have agreed upon the price of an item, and then another person comes and offers the buyer a similar item at a lower or better price, or offers the seller to buy his item at a higher price. This is because it causes enmity and hatred.