Frame challenge: both.
As can be seen in the other answers, the difference in damage output is relatively small in a vacuum. However, combat in D&D rarely takes place in a vacuum (and if it does, your bigger concern would probably be breathing). I see nothing in your situation that gives you a reason to permanently choose one or the other. I suggest you carry both. The various situations likely to be encountered by your barbarian will create more of a difference in your ability to kill foes than exists between the two styles in a vacuum.
Therefore, I suggest that the better question for your explained situation is under what situations should my barbarian use one style or the other?
Two weapon fighting is probably better if:
You are currently a small race. Disadvantage with heavy weapons makes them far less effective.
Low hit point enemies. If they die in one hit from either, killing three per round is far better than killing two per round.
The combat is likely to last a long time.
Damaging more than one enemy is desirable, such as attacking trolls with flaming weapons.
Great weapons are better if:
The combat is short, but you still want to rage, due to losing a bonus action attack the first round with two weapon fighting.
Bigger hits are desirable, such as against zombies.
You can obtain exactly one magical weapon for your character (perhaps one ally can cast the magic weapon spell) when fighting enemies with resistance or immunity to non-magical weapons.
You are currently a half-orc. (You said to ignore this, but this answer is a frame challenge.)
Your opponent has a very high AC, as a greater percentage of hits will be crits, thus favoring the larger die.
Composite Plating and Heavy Plating are intended to count as armor.
Keith Baker, original creator of Eberron and lead designer for the Wayfinder's Guide to Eberron, addresses this question (and many others) in the WGtE FAQ on his website:
Does the Warforged Heavy Plating Integrated Protection work with the Fighter’s Defense Fighting Style? Unclear on if (armor) means you’re “wearing armor” or not.
The intent is while you are using darkwood core you are not considered to be wearing armor for purposes of game effects such as Unarmored Defense, while you are considered to be wearing armor when you’re using one of the plating modes. However, as written Integrated Protection doesn’t allow you to benefit from the Defense Fighting Style. This won’t be changed until people have a chance to provide feedback on the race mechanics as they stand.
Baker points out that the feature doesn't specifically clarify that Composite Plating or Heavy Plating count as armor - but the parenthetical after each one reflects the intent, which is that Darkwood Core counts as being unarmored and that the other two options count as being armored. This is relevant for features like the Defense fighting style (which requires you to be armored), or the monk's Unarmored Movement feature (which, as the name implies, requires that you be unarmored).
In another response, he also says:
When a warforged is using the darkwood core (unarmored) mode and have light armor proficiency, can they choose whether they are considered to be armored or unarmored?
No. You are always considered to be unarmored when using darkwood core, and always considered to be armored when using plating.
So you can choose which of the three modes to use (depending on what armor you're proficient in), but you can't choose whether that mode counts as being armored.
...but as currently written, they don't count specifically as medium or heavy armor.
Baker also answers a related question about what kind of armor the Integrated Protection feature counts as:
Composite Plating and Heavy Plating both say “Armored” but do they count as medium and heavy armor respectively? Specifically does a warforged Barbarian in Heavy Plating mode lack access to the benefits of rage?
As written, the current version simply specifies that plating counts as armor, not that it counts specifically as heavy or medium armor. So in this version, a warforged barbarian can rage while in the heavy plating mode… though they will have to acquire the heavy armor proficiency before they can use heavy plating.
This may change in the future, but as of right now, a warforged barbarian can gain the benefits of Rage regardless of what defensive mode they're in. However, they can't benefit from features that forbid any armor at all while using Composite Plating or Heavy Plating.
Best Answer
No, a barbarian does not need to be in combat to rage.
Defining things as "Actions", "Bonus actions", "Reactions" etc. are only to specify how much you can do in a turn or round of combat, since you effectively only have a 6-second timeframe to do things. Much like casting spells that require a minute long casting time - you can still attempt to cast them... you just need to be uninterrupted for 10 rounds.
The only catch is that to sustain your rage, you need to attack (which doesn't have to be successful, mind you), or take damage, "every round", so a timeframe needs to be defined outside of combat to sustain it:
Note: the rule does state "hostile creature", so it'll be up to you and the GM to define what states as "hostile". Spanking a misbehaving child, or punching a passer-by may not exactly qualify.
However, as suggested in the comments, setting yourself on fire would be an ideal way to maintain the rage - constant damage, therefore, constant rage.