dnd 5e – Twin Arms Attack of the War Clerics

So one of my players is a war cleric and a doppelganger. He has carried out three attacks 1) the attack action 2) the bonus action for the other side and 3) one of his "War Priest Bonus Attacks" (PHB p. 63) for the third side.

But when I read it again, I just noticed that it is a bonus action.

So my question is, since this "bonus attack" is used as a bonus action and an "attack with the other hand" also uses his bonus action, does that mean that he gets only one or the other?

He specifically chose the Dual Wielder feat because he wanted to "cut and dice". I understand that the "War Domain Bonus Attack" is more for swords and boards or two-handed weapons, but it would be unreasonable to let him use this "War Domain Bonus Attack" (from which he gets 4) as the third one of the time attack at time?