Rage Beyond Death (XGtE 11):
You still must make death saving throws, and you suffer the normal
effects of taking damage while at 0 hit points. However, if you would
die due to failing death saving throws, you don't die until your rage
ends, and you die then only if you still have 0 hit points.
Death Ward has multiple triggers. The first trigger only kicks in when the barbarian takes damage that would drop them to 0 hit points for the first time. Since they are already at 0 hit points (There are no negative hit points, see PHB 196-197) this trigger does not occur at any point during Rage Beyond Death (unless the barbarian is healed, and thereafter drops to 0 hit points).
The first time the target would drop to 0 hit points as a result of
taking damage, the target instead drops to 1 hit point, and the spell
If the spell is still in effect when the target is subjected to an
effect that would kill it instantaneously without dealing damage, that
effect is instead negated against the target, and the spell ends.
The second trigger that negates any effect that would kill the barbarian instantaneously without dealing damage would negate the death that would occur after Rage Beyond Death when the barbarian is at 0 hit points at that time. If this trigger occurs, then the barbarian would only negate that instance of death, they would not get to 1 hit points.
That effect that is negated is the three death saving throws combined:
an effect that would kill it instantaneously without dealing damage
Singular (that instance):
that effect is instead negated against the target.
Effect means the result of a particular influence; the influence in question are the three death saving throws, once the third Death Saving Throws (PHB 197, my emphasis) occurs:
A success or failure has no effect by itself. On your third success,
you become stable [..] On your third failure, you die.
So all three fails are negated because they only produce the effect together that causes death. All successes remain.