SILVER SEAS:
General things to know: Ramparts call all reinforcements (usually multiple things at once) when aggro'd. Also note that ramparts aggro 15.0 at night, halved during the day and during dark/water weather. Ramparts don't drop alex. Sneak and Invis are recommended to skip past ramparts you don't need. Some of the moving ramparts can be aggro'd at the far end of their path to link fewer reinforcements. /DNC recommended for BST of course, and bringing a defensive shield is useful for Floor 3.
Floor 1: Just unlock Head/Legs/Feet and move on.
Floor 2: Skip everything if you got HP from the lamp. Otherwise, take a left and sneak/invis to avoid the 2 ramparts in the hallway, open the door on the left. The rampart there will summon a MNK, PLD, and BLM, kill them if you should feel HP is necessary (MNK gives it up). One of the later ramparts will also aggro true sight so just keep running.
Floor 3: Skip right to the ramparts that summon 3 single and one triple gears; you can kite these back to a previous room to deaggro a few gears, so you won't be so heavily outnumbered. Triple gears will unlock pet, singles will unlock the rest of your armor. These gears are moderately dangerous for a half-naked pet-less BST so playing defensively (shield, etc) is recommended until you can kill at least one. Pretty low alex payout on this floor, so you can go up when you've acquired your unlocks.
Floor 4: Immediately sneak/invis and head to the door, which will cause a rampart to spawn with a Dvergr sub-boss. Return to previous ramparts for alex farming if you want to, be careful to not link the ramparts (they will aggro/link if they are open) in the last 2 rooms or else they will call a Acrolith intead of a triple gear or chariot.
Floor 5: Dvergr boss, should go down very easily, just keep a safe distance because it has tremendous magic accuracy and fast cast.