Originally Posted by Cambridge dicts
used for emphasis, or to show that something is being done in opposition to someone else's wishes:
Mine's bigger than yours, so there!
No, I won't help you, so there!
Is it childish? Is it close to "and that's that!"?
Is it both AE and BE?
I think it's quite childish, yes.
I'm not a teacher, so please consider any advice I give in that context.