The battleground has been changing quiet a lot so I personally haven't tried A and B, but as of now battleground C is still only for Trans Characters, although they might change it soon too but for now it's basically just for Trans.
Battleground C still uses the old BG games, Tierra Gorge, Flavius and KVM.
Battleground B and C are currently Silk Mode. As Battleground A is normal mode (Sorry, I don't know the restrictions for normal mode)
Silk Mode: only items that are from the rental NPC are allowed to be used. No MVP Cards, KVM Weapon. Plainly just all he items that you can rent or if you already possess those items that are available in the Rental NPC, then you may use it.
Under the Silk Woe Guidelines, you can find the list of items that the rental npc offers, so are the cards that can be used in Silk setting. That list is mostly for 3rd jobs. Some gears might be different for the rental npc for Trans Characters as some gears does not work for Trans.
Someone kindly do correct me if I'm wrong. Thanks