How does pet insurance work for leagues?

The League Tutor, located in Kourend Castle Courtyard, will allow you to reclaim lost pets. Note that it is not clear whether this will cost 1 million gp.

From the wiki page about the Twisted League update (emphasis mine):

League-to-League Pets

Any insured pets gained in the previous League will carry over to the next one.

The League Tutor will allow you to reclaim your pets if you lose them.

Source

@Cookiezzz has shared with us that reclaiming the pet is, in fact, free! Any costs that are applied in the normal game, should not apply in Twisted League.

Just reclaimed a pet. It was free.