For a competitive couch co-op experience, you can't go wrong with "Mario Kart 8 Deluxe." The frantic racing and item-based chaos really bring out the rivalry, especially on those tight tracks. If you're up for a bit more strategic depth, "Overcooked! 2" can also twist the knife, working together seems easy until the chaos ensues, and competitive scoring can lead to heated moments.
If you're leaning towards something with a darker edge, "TowerFall Ascension" is a fantastic choice. The combat is fast-paced, and the maps are small enough that it feels intensely competitive, especially with the right item placements. Have you guys tried any of these, or do you have favorites that might influence what else I suggest?