You can make unarmed strikes while Wild Shaped, but they do normal unarmed strike damage.

From the errata for the PHB:

Melee Attacks (p. 195). The rule on unarmed strikes should read as follows: “Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons). On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier. You are proficient with your unarmed strikes.”

An unarmed strike is now defined extremely clearly. On the plus side, an unarmed strike can replace any attack with a weapon. From the Monster Manual, page 10:

The most common actions that a monster will take in combat are melee and ranged attacks. These can be spell attacks or weapon attacks, where the "weapon" might be a manufactured item or a natural weapon, such as a claw or tail spike.

So your unarmed strikes can definitely replace your natural attacks.

The downside to this is that your natural attacks definitely don't count as unarmed strikes, so you can only do your normal unarmed strike damage. This was also confirmed by Jeremy Crawford.

Only a little, if at all.

A +3 Spiked Armor that applies enhancement to Battlerager features essentially serves a similar function of a Defender that always activates (since you only get one attack). So we can use the differences to measure balance.

The first time you attack with the sword on each of your turns, you can transfer some or all of the sword's bonus to your armor class, instead of using the bonus on any attacks that turn.

Not being able to use the weapon bonus on multiple attacks is a significant downside, but you also get the benefit from the other Battlerager features.

Here is a cost-benefit analysis of the tradeoff for comparison.

• Damage per round: 3 damage per Extra Attack vs 3 damage per attack that hits you.
  • I would say these are comparable if not in the favor of Defender as smart monsters will avoid attacking you in melee (most have a ranges or magic option at the level where you usually start getting magic armor).
• Armor bonus: 3 if you sacrifice your extra damage vs 3 always
  • This shows the strength of the variant magic spiked armor. You have a consistent armor benefit. This is where the slight imbalance comes from, but it is probably not enough to unbalance the game substantially.
• Grapple damage: Grappling in 5e is very niche but the 3 extra damage does confer some small benefit compared to the Defender.

None of these small differences would push the weapon above or below the power level of some of the other legendary weapons (of which the Defender is included). While it is not a direct comparison for the lesser magic armors (with +1 and +2), the principle can be extrapolated to include them.

If you find that this is unbalanced since it is bonus damage for a non-attunement item, You could make the Battlerager benefits require attunement, but I doubt it will be necessary.