Is there any move that cures the enemy pokemon's status problems?

Solution 1:

There are at least three ways. Unfortunately, none will work on your scenario.

Fling, when used with a corresponding status healing berry, should recover the status condition. However, it counts as a 10 Power attack prior to causing the effect, so you would not even be able to throw a Sitrus berry for healing before you would kill something at 1 HP.

Trick and all of its derivatives can be used to swap a status healing berry onto the target. None of the Legendary Beasts have held items to my knowledge, though, so this won't work.

Finally, the moves Wake-Up Slap and Smelling Salts will cure Sleep and Paralysis respectively. But not only do they do damage, they do double damage to such an afflicted Pokémon. So like Fling, not viable.

Solution 2:

Kill it. No, really. After you defeat the Elite Four again, any roaming legendaries you knocked out will return to the the Brass Tower, at full health and status effects.

As for capturing Entei, beware that roaming legendaries can flee even when asleep. You need to use Mean Look or Block, or a Pokemon with the ability Shadow Tag or Arena Trap. This will give you more than one round to throw Pokeballs, and greatly increase your chance of capturing Entei.

Entei can still escape using Roar, but blocking this while keeping a no-fleeing effect in play involves a lot of specialist breeding work. The easiest is to level an Eevee to L36 to learn Baton Pass, evolve it to an Umbreon and level it to L57 to learn Mean Look. Use this to trap Entei and Baton Pass out to something with the ability Soundproof or Suction Cups or the move Ingrain, any of which will block Roar. Many such Pokemon can learn a sleep or paralysis move by breeding or level.

Both Entei and Raikou are fast enough to benefit from Fast Ball, which gives a 4x catch rate, better than an Ultra Ball. There is no better Pokeball for Entei or Raikou.

Solution 3:

Use Manaphy to Skill Swap its Hydration ability to the target pokemon. Since you only have one turn, you may have to do this on route 33 where it rains naturally.

Unfortunately, no pokemon with Leaf Guard or Shed Skin (my first idea) will learn Skill Swap.