Does Dusk Rose Lab apply first or Temporary Units Dying apply first?

At the start of your Beginning Phase, both the Temporary trigger and the Dusk Rose Lab ability are added to the chain. Because you control both, you choose the order in which they are placed on the chain, and therefore, the order in which they resolve.

According to RiftJudge FAQ #9428 and #9231, you can choose to resolve the Dusk Rose Lab ability first.

Sequence of Events

1. Beginning Phase starts: Both the Temporary trigger (which aims to kill your temporary unit) and the Dusk Rose Lab trigger are added to the chain.
2. Chain Resolution: As the controller of both, you choose to resolve the Dusk Rose Lab ability first.
3. Cost Payment: You kill your temporary unit as the cost for the Lab's ability.
4. Effect: You draw 1 card.
5. Temporary Trigger: The Temporary trigger then attempts to resolve. Since the temporary unit is no longer on the board (it was already killed to pay for the Lab), the instruction to kill it cannot be followed and is ignored.

This interaction is confirmed by RiftJudge FAQ #9231, which explicitly states that you can use the ability of Dusk Rose Lab to kill a temporary unit before it is removed by its Temporary trigger.