Can you use the ability of dusk rose lab to kill a sprite before it dies to draw?

Yes, you can use the ability of Dusk Rose Lab to kill a Sprite token before it is removed by its Temporary trigger to draw a card.

Reasoning

The Temporary keyword on a Sprite token creates a triggered ability that goes on the chain at the start of your Beginning Phase. Because this trigger uses the chain, you have a window to react to it before the Sprite token is removed by the Temporary effect.

Since Dusk Rose Lab also triggers at the start of your Beginning Phase, you can choose to resolve the Lab's ability first. By using the Lab's ability to kill the Sprite token, you fulfill the cost of the Lab's ability and draw a card. Because the Sprite token is already killed by the Lab's ability, the Temporary trigger will fail to find its target when it attempts to resolve.

Sequence

1. Beginning Phase starts: Both the Temporary trigger (from the Sprite) and the Dusk Rose Lab trigger are added to the chain.
2. Chain Resolution: You choose to resolve the Dusk Rose Lab ability first.
3. Cost Payment: You kill the Sprite token as the cost for the Lab's ability.
4. Effect: You draw 1 card.
5. Temporary Trigger: The Temporary trigger attempts to resolve, but since the Sprite token is no longer on the board, the instruction to kill it cannot be followed and is ignored.

Note: I could not fully verify this ruling against the retrieved rules.