The pets scale with Ilvl, I just did a +6 last night as Resto Shammy/4x Hunter and completed it in a +2, was decently close to +3.
A neat trick I have as part of my spam is round robin Growl/taunt rotation, it's on a 8 second cooldown, so you could have it cycle every 1.6 seconds. That way each pet has time to actually heal in the process.
I experience the opposite issue where some trash demolishes the pets but the bosses don't do much of anything. Which is mostly due to having 6-8 mobs aoeing down.
My ilvl is 370-382ish right now on this team, so I still have quite a bit ways to work up.
Having a Shaman spamming chain heal (which bounces to pets as well) really helps keep them topped off.
So if you get frustrated with no heals, remember there is the option of adding a healer to the mix and being just as effective! Also wears mail so gear sharing is easy.

To boost your ilvl up, do whichever warfront is up, the arathi one gives out ilvl 340 for completion every time and the first time also gives a 370, and the darkshore is 355 and gives a 400 item on the first time of the cycle. Then do emissaries if they give a rewards, and world quests should be boosting it up quite a bit. They will end up scaling all the way to 385 for the emissary reward.