The second benefit of the Dual Wielder feat (PHB, p. 165) refers to the "Two-Weapon Fighting" section of the PHB found on page 195:
When you take the Attack action and attack with a light melee weapon
that you're holding in one hand, you can use a bonus action to attack
with a different light melee weapon that you're holding in the other
hand. You don't add your ability modifier to the damage of the bonus
attack, unless that modifier is negative.
If either weapon has the thrown property, you can throw the weapon,
instead of making a melee attack with it.
As such, the second attack requires a bonus action, as described in the Two-Weapon Fighting rule.
Yes, you can use War Priest but not if you use Multiattack
Taking a cue from András's answer, the attacks listed in a creature's stat block are not Actions but attack modes or melee attacks. Here's what I think you're missing:
Page 192 of the PHB defines the Attack Action:
Attack
With this action, you make one melee or ranged attack.
The PHB further defines what a melee attack is and, in page 195, it explicitly states that:
Melee Attacks
A typical monster makes a melee attack when it strikes with its claws, horns, teeth, tentacles, or other body part.
With the above reading, the PHB confirms that Claw, Gore, Bite, and Slam are melee attack options that can be chosen when a creature uses the Attack Action and therefore able to trigger War Priest.
But this is where it gets weird:
Multiattack is a distinct Action that details what attack options are used by the creature when the creature uses Multiattack (not the Attack Action!). Since it is a separate Action (like Ink Cloud and Swallow) it cannot trigger War Priest.
This means creatures using Multiattack cannot trigger War Priest; they must use the regular Attack action and then use a Bonus Action to attack with War Priest, but this obviates the need to use War Priest as the Druid/Cleric can already attack twice with Multiattack.
As pointed out by KorvinStarmast, a beast with different attack options in the Multiattack Action like the Brown Bear could potentially get a slight damage increase if the Druid chooses not to use Multiattack (Claw [2d6+4] + Bite [1d8+4]) but instead use War Priest to attack twice with Claw (4d6+8). This allows for a unique dynamic that a Druid/Cleric can use during combat, when needed.
Best Answer
Correct
You only get one bonus action to use on your turn. You can't make a dual-wield offhand attack and use the war priest ability in the same turn.
As for unbalanced? If they were using a two-handed weapon they'd get the most damage from a greatsword or maul which would do 2d6 + strength modifier per hit. So if they hit with both attacks they'd do 4d6 + 2 × strength modifier per turn.
For dual wielding the best you can do, given the dual wielder feat, is a pair of 1d8 + strength damage weapons. That would give you 2d8 + strength modifier × 2 max damage per turn.
Now if you consider the combo you are suggesting you could do 3d8 + strength modifier × 2 max damage per round. That is slightly less average damage per turn than the two handed combo given by a greatsword/maul and war priest attack.
In terms of damage then it isn't really unbalanced. The number of attack rolls is a slightly different matter. One more attack roll per turn increases the odds of a critical hit on each turn slightly. Instead of two attacks, each with a 1 in 20 chance of a critical (assuming nothing else is affecting the range of values), you have three attacks with the same odds. Not huge, but significant. It also increases the odds of a critical miss but you are pretty much never going to hit on a 1 anyway so it doesn't really a difference.