What you were expecting to happen, and what actually happened:
Team Score to be the same with Hero in Pet Rescue as it is in Team Troop tab
Team score in Pet Rescue with Hero Not the same
From what I can tell your team score is based mainly on your kingdom bonus skills, level of each troop on the team, and number of unlocked traits for each of those troops. Higher team score doesn’t mean you can’t beat it, but it’s a snapshot of how much harder the team may be.
Going by your screenshots, your Pet Rescue team doesn’t have a hero class assigned. Does it change your score if you set it to the Warpriest class your other team uses?