Can I react to Spinning Axe's temporary with Angle Shot to protect it?

No, you cannot save Spinning Axe by reacting to its [Temporary] trigger with Angle Shot.

According to the official FAQ regarding this interaction:

• Ruling: Yes, the Spinning Axe will still die. When you respond to the trigger with Angle Shot, the trigger is already on the chain, so it will resolve and destroy the Spinning Axe even though it has been moved to a different unit.

Sequence:

1. Spinning Axe's [Temporary] trigger goes on the chain at the start of your Beginning Phase.
2. You respond with Angle Shot, moving the Spinning Axe to a new unit.
3. Angle Shot resolves.
4. Spinning Axe's [Temporary] trigger resolves, destroying it on its new unit.

As noted in the rules, [Temporary] triggers at the start of your Beginning Phase. Because the trigger is already on the chain, changing the attachment status of the gear after the trigger has been placed does not prevent the trigger from resolving and destroying the card.