Yes, a Battle Master can give up their single-attack Attack action to use Commander's Strike.
You appear to be confusing the Attack action with an attack.
The Attack action is one of the things you can choose to do with your action; others include the Dash action, the Cast a Spell action, and the Use an Object action.
An attack is anything you do that causes you to roll a die to overcome someone's AC. Trying to hit with a weapon is an attack, but so is using a spell that rolls against AC, such as eldritch blast. You can make attacks without taking the Attack action, e.g. attacking with a spell using the Cast a Spell action, attacking with a magic item using its own unique action, or attack with a weapon using a reaction (e.g. using the Ready action, or making an opportunity attack) or bonus action (e.g. from Two-Weapon Fighting).
The prerequisites for the Battle Master fighter's Commander's Strike maneuver are that you:
- take the Attack action, which always allows at least one attack, and
- forego one of those attacks, even if it is the only one
The rules for grappling and shoving a creature are similar.
This is not balanced with the other races in the PHB.
Assuming that these are two different sub races ...
- You are starting with a feat.
- You are starting at size large: no PHB race is large. (Not even Goliath)
Why this may matter:
- Weapon damage changes. You get 1d4 bonus when you go from medium to large (see potion of growth, enlargement spell). There is a reason all player playable races are medium sized. The PHB doesn't explain how damage goes up for a Large sized PC because No PC is sized Large. It's a default.
- Monsters get additional damage dice if large. (This should not apply to you, though).
- Being large also allows you to Grapple and Shove Huge sized opponents. It also means that small sized monsters can't grapple or shove you (Kobolds, Goblins). That becomes a non-trivial benefit as you fight things larger than large at higher levels. You can ask my half-orc champion versus giants, cyclops, etc, on that score. Without a potion/spell, his shield master shove does not function.
- Unarmored AC is overly generous, given that class isn't yet given.
Natural weapons at 1d6 + str ... significantly more than PHB races. (Though similar to Tabaxi from Volo's ... 1d4)
The base class benefits of climb, cold resistance, etc fit well enough on their own for the base race. You added a further two skills in the searching for perfection. (Equivalent to half elf skill boost).
Your Proposed bonuses:
For Eir: Dex +2. With two feats. Similar to vHuman.
For Hiojuth: Str and Con + 1. Similar to vHuman.
Compare to variant Human: +1 to two abilities and one feat, and one added skill.
Skills: Your "nimbles claws" as thieves tools is an equivalent to a vhuman added skill, but it isn't a skill. Using it as a bonus action begins to approach a feat. Add to that unarmored defense and you are out of balance.
Add Athletics Proficiency, which makes for three additional skills for the Hiojuth sub-race, isn't too far out of whack. You then added the half-orc racial unique ability for Hiojuth on top of that. And then unarmored defense ...
Natural Armor: natural armor (6 when un armored, ac = 10 + nat armor + dex mod)(maybe changing that to con mod?)
Natural armor as presented is equivalent to roughly plate armor with no shield: 10 + 6 + either Dex or Con mod. (Depends in how high your dexterity is at roll up, to which you have already added 2). If your starting Dex is 16, you have AC 19 without armor. A monk with Wis 16 and Dex 16 has a 16 AC. If you picked a monk, and you have a decent wisdom (14) and 16 dex (Eir), your unarmored AC is 20.
No, this is not balanced.
What to fix?
Size: Medium. Like all other races in PHB and Volo's. (Except for the small ones ...)
Base race: Good! But change Natural Weapons to 1d4 + bonus
Sub races: Lose the feats. Keep the rest.
Eir: Make nimble claws "equivalent to thieves tools" but drop the bonus action.
Unarmored AC: lose it, or trade one of the other base class attributes for a +3 bonus.
Notes.
The unarmored attribute is usually a class ability (Draconic Bloodline sorcerer, Barbarian, Monk, although a few monster races like Tortles or Lizardfolk have something like this).
An argument for "lose it" comes from @Ethan, who made a point in comments about how the game treats Armor Class, and bounded accuracy: the unarmored defense pushes up against a 5E design principle. There are almost no plain AC bonuses. Typically, the game presents an alternate calculation like similar to mage armor. "While not wearing armor, your AC is 12(or 13) + dex mod" would fit the design model more closely. (See the Sorcerer example in the PHB).
Best Answer
Your unarmed strikes do not gain mechanical benefit from extra hands
Fluff-wise, there is no constraint regarding what your unarmed attacks actually represent. It could be punches, chops, kicks, headbutts, things like that. However, a single unarmed attack deals the same damage as usual, and in order to do multiple attacks, you need features like the Monk's Martial Arts or Flurry of Blows, or normal extra attacks, to get more attacks per turn. If we were meant to assume the homebrew race is supposed to grant extra attacks or augment the damage of unarmed strikes, the description of the homebrew race's feature should say so.1
Note that extra free hands do not grant extra attacks or unarmed strike damage to anyone else, either. A character with two free hands deals the same damage as an otherwise identical character who has no free hands and opts to kick their enemies instead.
1 A little footnote pertaining to the racial feature's wording: a literal reading of the racial feature makes the race LESS capable of multiple attacks per turn than other characters. The phrasing "you can only use 1 [weapon] per turn" disallows Two-Weapon Fighting (or spreading extra attacks between different weapons), the first clause disallows multiple attacks from Martial Arts, Flurry of Blows, and any other source that isn't "Multiattack" (presumably meaning the "Extra Attack" class feature), and a very pedantic reader would also find issue with "You can only attack once" (as opposed to "...once per turn" or a similar wording).