Mirkwood is in fact a SUBFACTION, not an actual independent faction.
Furthermore, its power and role in the 'geopolitics' of the War of the Ring are definitely inferior to Lothlórien's ones.
Man, therefore ... there would be great an idea to make Mirkwood as own faction. Lorien and Mirkwood are two different realms
They are indeed two different and isolated realms in the War of the Ring.
But, we are still talking about the same Elven Kin, mainly Sindar, Nandor and Avari (Galadriel is the only exception).
Both the Lords of Lothlórien and Mirkwood (Celeborn and Thranduil) have their background in the Sindarin Royal Court of Doriath in Beleriand, during the War of the Jewels and before.
Given these elements, it won't nonetheless be possible to have a new Elven faction just for Mirkwood, due to the very Edain Mod's mechanics and structure (number of heroes, structures, spell...).
Not even the Humans have more than three factions, for now; why should the Elves, who are diminishing and leaving the World in the Late Third Age, be represented with another third faction?