One additional approach, which might be more closely aligned with the available tools in Spatial Analyst is to reclassify the landcover classification in such a way that a focal sum could create unique outputs based on the values within the neighborhood window
e.g. quadrant1 (3 x 3 grid) = 100,000 + 10,000 + 1,000 + 100 + 10 + 1 = 111,111
quadrant2 = 10,000 + 10,000 + 1,000 + 100 + 100 + 10 = 21,210
such an approach would give both what amounts to a list of values and the relative abundance of each class.
From there, you could simply calculate a second focal stats using 'variety"...