How about Advergame?
(Advergames, in the context of video game genres, refers to promotional software specifically made to advertise a product, organization or viewpoint. The first advergames were distributed on floppy disk by the Chef Boyardee, Coca-Cola, and Samsung brands,[32] while the first cereal box advergame was Chex Quest in 1996.[33] The majority of advergames are found online and mostly include simple and cheaply made Flash games.)
MOBA is a terrible subgenre name. Sounds like a Roomba.