Firstly, Double Team is banned in competitive play. Replace it with Foul Play.
Give your Umbreon this set:
Heal Bell/ Confuse Ray
Secondly, if your opponent sets up Swords Dance, they won't be using a special move like Flamethrower...
In any case, I would use Confuse Ray first, so it has a chance of hitting itself and you have a chance to take less damage, more time to set up Toxic, and then heal yourself with Moonlight. If your opponent switches Pokemon, use Toxic then (if it's a Pokemon you're confident Umbreon can wall). Then again, if they know Umbreon has Toxic, they'll probably either switch to a cleric like Blissey or a steel type.
Foul Play is the best offensive move for Umbreon. It will do damage calculated on your opponent's attack, not yours, and since Umbreon has abysmal attack, Foul Play is perfectly suited for it to get rid of physical sweepers (that are non-fighting type).
You also have to consider that the opponent won't switch out, probably due to the fact that they've already set up a Swords Dance and will take their chances, and that's when you use Foul Play to get rid of a Physical sweeper.