Is offering lower prices to ensure winning a bid for selling computers, while the center is aware of other offers from the same entity, included in the Prophetic prohibition, "And let not some of you sell upon the sale of others"?

This does not fall under the category of selling over another's sale. Rather, it falls under the permissible bidding mentioned by the Prophet (peace be upon him) in the Hadith (definition) of Tirmidhi. As for trying to find out the prices of others in order to offer a lower price, if it is without spying, there is no harm in it. But if it involves spying, then it is not permissible, due to Allah’s saying: "And do not spy."