Trying to tank isn't a good option for this fight, all three of his pets have high dps that will overwhelm just about any tanking pet.
The bear opens with either rampage or bash: if he bashes, just concede and restart the fight before you take any damage.
Things that kill the bear easily: Darkmoon Zeppelin, Grunty, Baby Ape, Darkmoon Tonk. I've tried this fight using a flyer to try to tank the damage and it isn't nearly as effective, because a flyer can't do enough damage and the bear just heals and gets another 3 shots of Rampage on you: you want a high damage mechanical attack to kill it before it finishes the first Hibernate. You should also be able to do the requisite damage with a burst ability like Ghostly Bite/Surge of Power, but this is most likely going to get your first pet killed vs the Mechapeep right after.
Most elementals kill the Mechapeep easily, I use Phoenix Hatchling if I want the easiest time of it. Again you don't want to tank, you want to rush it down, so Lil Ragnaros and Fel Flame are especially good, but there's lots of solid choices here.
The Lasher is probably the easiest as long as you keep pressure on it. It will mostly choose to heal itself as long as it's low on health, which basically eliminates damage on your guy. Most aquatic pets can handle this: I really like the ones with Pump and a heal (Mirror Strider, Purple Puffer, etc.). If you try to tank it though, it will Lash for 500+ per turn in addition to Entangling Roots. This is not impossible with Magical Crawdad or a Snail/Whelk (e.g. Rapana Whelk in Dread Wastes), but I think it's the harder route.