Can a money service business cash a business check?

Yes they can if they equip themselves to offer such a service. The general definition of an MSB (Money Service Business) by FinCEN is:

The term "money services business" includes any person doing business, whether or not on a regular basis or as an organized business concern, in one or more of the following capacities:

(1) Currency dealer or exchanger.

(2) Check casher.

(3) Issuer of traveler's checks, money orders or stored value.

(4) Seller or redeemer of traveler's checks, money orders or stored value.

(5) Money transmitter.

(6) U.S. Postal Service.

No activity threshold applies to the definition of money transmitter. Thus, a person who engages as a business in the transfer of funds is an MSB as a money transmitter.