Easier Way To Get One Of The Legendary Dogs:

Have you already used the MasterBall on Mewtwo or on one of the Legendary Birdies? Then there is a way to get the Legendary Dog based on your Starter. Catch a Wobbuffet in the Cerulean Cave around Lvl 58 or higher. You should find one if you go deeper where the levels are higher on that floor.

Once if you have a Wobbuffet, Put him in the first Slot and look for the dog. If you already have seen him. then you must look for the nest in the Pokedex. The dog will change from area if you reach a new area/walk into a building/or Fly to another city, be carefull.

If you are on the same area as the dog, don't leave the area but just look for him in the grass. The Dog will appear in a battle. if you notice, you'll hear a special Wild Pokemon Battle Music. If Wobbuffet appear in battle. the dog can't flee, cause Wobbuffet's Shadow Tag Abbility makes Wild Pokemon or Trainers unable to switch or run. And that's your chance to weaken the dog, and toss Balls, make sure you have major balls around the 100, and enough Potions to heal obbuffet,
cause the dog will fight instead of running.