The MEWTWO would have something like
ICE BEAM/THUNDERBOLT (else, why CHANSEY)
In this case, it might be worth ditching LIGHTSCREEN for TOXIC and relying on SOFTBOILED to survive while accumulating DOUBLE TEAM effects, or ditching DT and relying on LS (though that's specific to MEWTWO, and so less flexible). In any event, the MEWTWO would still suffer from a missing elemental attack, so LAPRAS or JOLTEON could fill in if CHANSEY became exhausted as needed.
LAPRAS and JOLTEON aren't specifically required, btw. Any strong ICE+FIRE and ELECTRIC resistant pokeymans with decent special and no psychic weakness would do.
Also, on the subject of anime, why the fuck is there a normal dog in this scene?