Does an Arcane Armor replace the legs of the Centaur

Unfortunately there's no official rule that says you can do this.

In a home game, it would be fine. You'd talk to your DM and explain that you can't wear this armor as a barbarian, but that you'd like to just wear some of it and count is as half plate. Most reasonable DMs would issue a house rule saying you could wear some of the armor, just as you're suggesting.

However in Adventurers League you can't really do that. One version of the AL DM rules says:

You’re Empowered. Make decisions about how the group interacts with the adventure; adjust or improvise but maintain the adventure’s spirit. This doesn’t allow you to implement new rules, however.

So the DM is not allowed to let you do this.

I recommend trading the armor away to a character that wears heavy armor.

It's up to the DM to use common sense.

"Use common sense" is exactly the guidance given in the Dungeon Master's Guide (p. 141), under "Multiple Items of the Same Kind":

Use common sense to determine whether more than one of a given kind of magic item can be worn. A character can’t normally wear more than one pair of footwear, one pair of gloves or gauntlets, one pair of bracers, one suit of armor, one item of headwear, and one cloak. You can make exceptions; a character might be able to wear a circlet under a helmet, for example, or be able to layer two cloaks.

The Armorer Artificer's Arcane Armor feature says, in part (TCoE, p. 15):

The armor attaches to you and can’t be removed against your will. It also expands to cover your entire body, although you can retract or deploy the helmet as a bonus action. The armor replaces any missing limbs, functioning identically to a limb it replaces.

To this DM, it looks like you would be functionally wearing boots and gloves, since the armor covers the whole body. As for a helmet, it seems there should be no issue wearing another helmet while your Arcane Armor helmet is retracted, just be careful not to engage your helmet while wearing another one.