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I like the idea of a Triple-Double Battle. The name is misleading, but I couldn't resist.
Basically you'd group your 6 pokemon into 3 teams of two, and then have 3 consecutive battles of your team vs. the opponent's. Whoever wins 2/3 wins the match. If you won the first battle with both your pokes still fine, and then lost the second and third, the fact that your first team was still fine wouldn't matter.
Could be neat. Or alternatively, you could have it that it's a fight till all 6 of one person's pokemon have fainted, only you'd still group your pokemon into teams of two and you could only have members of a team out at the same time. Switching would mean switching both pokes out at once for another team of two, and if one of your pokemon fainted, another wouldn't immediately take its place. Instead, a new team of two would take its place once the second mon fainted as well. Or you could switch to another two-poke team.
I'm very sleep deprived, and there are a lot of flaws to this idea. But I still like the idea of trying to create synergy between two individual pokemon.