It’s confusing because in interviews the priest is saying he was asked to sign it, but is leaving out that he already did.
But if you read the whole article, you’d have seen this bit buried below
The form, which Hood signed in order to attend Smith’s execution, gave an overview on the risk of nitrogen gas. It stated that in the “highly unlikely event that the hose supplying breathing gas to the mask were to detach, an area of free-flowing nitrogen gas could result, creating a small area of risk (approximately two feet) from the outflow.”
If he hasn’t signed it, they could proceed without him.
So he signed it, and now the lawsuit is happening to stall the execution.