Possibly. Level 2 and 3 interchange rates are discounted rates offered by the card issuers when you send up more qualifying data about the businesses and items processed for a B2B transaction, for qualified cards only (Corporate, Government, Purchasing, and some Business cards).
How do they know if it's B2B? Well, the card itself is issued as an eligle card first. If it isn't a qualifying card, then even if you send data, the issuer considers it a B2C transaction and you pay the normal transaction fee. IF the card qualifies as eligible, then the card issuer will provide a discounted transaction rate if you pass all the information they require with the transaction. One note, the connected gateway must also be configured to pass the data through to the processor.
Talk to your Sales Rep about Level 2 and 3 pre-qualifications.