I'm planning on playing Tyrant in an upcoming tournament, and I'm pretty sure he's going to be going up against a power battery or two, so I wanted some input on how his trait specifically works against the battery. His trait reads,

Adjacent opposing characters can't use relics or resources. When Tyrant is adjacent to an opposing character assigned a relic or resource, modify his combat values by +1.

So the part I'm not sure about is whether adjacent characters can still use constructs. Constructs from the battery aren't relics or resources, but they came from a resource. Any thoughts?

I would rule that the characters that can start with their on constructs (or things like DP's word bubbles) work just fine. However, characters that got their constructs from a battery would not be able to use them as they are part of using that resource.

Well, if a character can't use the portion of the Power Battery that says "can use the ability of Constructs on its card", then it can't use the Constructs despite them being on the card. Yes, Constructs aren't relics or resources, but without something telling you what having a Construct on your card does for you, it doesn't actually do anything.

They will still qualify as a character with a Construct on its card.

Interesting. That makes him a little more exciting than I thought, and I was already pretty pumped about him. Thank you both!