Well I finally managed to get Anzu down as Moonkin spec, and I have to say it is a lot harder as Moonkin then Bear.
Seeing Morrid on the forums downing Anzu as Moonkin inspired me to try this again, and following some of the tips from Moomuerto I adjusted my strat and tried it again, finally I got Anzu down after over an hour of trying and 100g in repairs.
I tried again a couple days later, and it still took me a few attempts, but this time only 30 minutes and 50g repairs, hopefully I'll be able to one shot it soon enough.
Here are some things I have taken away from my many attempts as I tried different gear, and I tried between my 2 Moonkin specs (1 raid moonkin, the other is a restokin pvp spec)
Stand against the wall as Moomuerto said at the start, hot the falcon spirit totem to increase your dps, then thrown a regrowth and rejuv on yourself and go to town on Anzu.
I found i usually opened with a starfire, moonfire, faerie fire, and wrathed till eclipse. Then proc eclipse and Anzu should faze out as he reaches 66%.
Run across to the other side of the room, thrown hots on yourself and get ready for the birds, make sure you get into Moonkin form before the birds swarm you as you get your heavy armor and 15% damage reduction while stunned. Anzu occasionally stuns you while he's phased out, wait out the stun, then hurricane the birds down.
Unfortunately my raid spec I don't have gale winds so it takes me 2 x hurricane, to get them down, with them bashing on you it may reduce the effectiveness of your hurricane due to pushback so you may have to do this twice
I generally find it easy getting past this first phase, but because my raid spec has no mana regen talents, I need to pop my innervate at this point.
Anzu comes out of his phased mode when all birds are down or 1 minute passes.
I've found it to be effective if only a few birds remain up, and Anzu still has some time left in phased out mode to shift out and throw on your regrowth and rejuv hots, since the birds won't hit you for much and Anzu hits like a truck when your in caster.
Finish the birds then dps Anzu again, once again I find it all about proccing eclipse to get him down quickly before your hots run out, if you spend too much trying to kite and dot your just wasting valuable dps time and your heals won't keep up, I throw the moonfire up as i have both the moonfire idol and t9 bonus, if I didn't i would probably just wrath to proc starfire (or the other way around) as this brings him down fast.
Enter the last bird phase and kite over and heal up again, same strategy as before, when Anzu phases back in burn all your CD's and finish him off.
I usually save my DPS CD's because it's much more heart breaking to fail when he's at 10k HP, then at the start. Also they give you a bit of leeway if you do make a mistake heading into the last bird phase or don't get lucky procs for eclipse.
The trinkets I've found the most helpful are my 2 PVP trinkets, battlemaster's bravery and titan forged rune of audacity.
I save them for the last phase to get out of a stun and to give me that extra 3k life boost to finish him off.
Due to my raid spec I find mana an issue so I do use a mana pot on my non fail attempts (ok sometimes i take a pot and still fail :( )
If you find your falling behind heading into the bird phase, or eclipse just won't proc at the start and you know your losing the DPS race, you can kite out of the room to reset Anzu, this saves you a repair bill, and the long run back.
If your dying a lot, it is worth clearing out the adds in the first hallway, and in the middle of the first room as you enter the instance to increase your running back speed.
I don't bother hotting the totems besides the start as I find i just don't have the time (or mana)
Removing the curse is the most important aspect of this fight, not only does the curse do significant damage, but the GCD knockdown and mana it burns really set me back.
I was wearing 2 x T8 and 2 x T9 for my attempts with PVP off pieces and gear in my other slots to make me as tankie as possible without losing too much dps.
In the end i ran with around 21.5k HP, 2.6k SP in Moonkin.