If I hide a card like Smoke & Mirrors at Battlefield B with a Sprite token at that Battlefield, then move a Lillia to Battlefield A and start Showdown, am I allowed to react with Smoke & Mirrors to swap the Sprite at Battlefield B with my Lillia?

Yes, you can do this.

Here's why: FAQ #10085 directly confirms that Smoke and Mirrors played from hidden can target a unit at the hidden battlefield (B) and another unit at a different location (A), because the card's requirement of "another unit at a different location" makes it impossible to fulfill using only targets at the hidden battlefield — an exception to the normal hidden targeting restriction.

On timing: Hidden cards gain Reaction (Rule 811.6), and FAQ #7354 confirms you can play a hidden card from one battlefield during a Showdown at a different battlefield. Since the Showdown at A is in Open State (before any chain resolves), you have Focus as the Attacker (since Lillia applied Contested), and Smoke and Mirrors has Action — you can play it.

The Sprite has Temporary, so the swap condition is satisfied. Lillia moves to B, the Sprite moves to A, and the Showdown at A continues with the Sprite there instead. This is consistent with FAQ #10888 and #11212 which cover the same Lillia + Smoke and Mirrors swap pattern during a Showdown.

One note: You must choose one unit at B (where Smoke and Mirrors is hidden) and one at a different location (A) — you cannot swap two units at locations neither of which is the hidden battlefield.