This is the full list of Ability-affecting Abilities I've found, I will update my answer if I find any others. There's a link in the comments that should give you Stat-affecting Abilities.
Abilities that affect the opponent's Ability
- Neutralizing Gas suppresses all the Abilities on the field (Both Weezing lines)
- Lingering Aroma changes the attacker's Ability to Lingering Aroma when hit with a contact move (Oinkologne)
- Mummy changes the attacker's Ability to Mummy when hit with a contact move (Unovian Yamask line)
- Wandering Spirit swaps Abilities with the attacker when hit with a contact move (Galarian Yamask line)
Abilities that change the user's Ability (Though these don't change opposing Abilities they can occasionally pressure a switch)
- Wandering Spirit swaps Abilities with the attacker when hit with a contact move (Galarian Yamask line)
- Imposter turns the user into the target and copies its Ability as well (Ditto)
- Receiver copies the Ability of a fallen ally in Double Battles/Triples (Passimian)
- Trace copies your foe's Ability (the Pokemon directly in front of it) (Alakazam-M, Porygon, Porygon2, Gardevoir line)
Overall, the Pokemon you should bring depends on the format, but I'm happy to help!
Source: Bulbapedia (https://bulbapedia.bulbagarden.net/wiki/Category:Ability-changing_Abilities)
