The player can leave the ruins and the count will not reset. Carding does count as defeating the Tonberry for the purposes of summoning the King, but Tonberry King will not appear in a battle where the Tonberry is defeated by carding. Tonberries killed in Ultimecia Castle also count toward this threshold, but the King can never appear there.
Tonberries will counter most attacks with Everyone's Grudge, an attack that deals HP damage calculated by the number of enemies the character has killed multiplied by 20. When it walks forward at least five times, the Tonberry will randomly attack a party member with its knife or it might fall over, losing a turn.
As the Tonberry's HP and defense are formidable, one may consider using Demi or the summon Diablos, as it is not immune to the Gravity element. Doomtrain or Meltdown can cast Vit 0 on the Tonberry to knock down its defense and make it easier to defeat. Other defense-ignoring attacks include Quistis's Micro Missiles and Irvine'sShot using AP Ammo. Keeping Irvine in low health and using his Limit Break is one of the fastest ways to amass the required kills to summon Tonberry King.
MuggingChef's Knives is a quick and easy way to obtain 100 Deaths for every character as one refines into 30 Deaths with Siren's L-Mag refine. Chef's Knives also refine into AP Ammo, useful for using the low-health Irvine tactic against them.