Just like to say I have yet to do Garuda hard so I am attacking this problem mechanically not from experience.

Ok so the hate stack is effected heavily by regen (possible bug not sure havent checked starting to think its front loaded so the whm gains all the enmity of the potential heals immedately as one with the buff enters the enmity list) so once that inital hit puts you on the listing try provoking to match you with the healers emnity then combo to butcher's and with a little luck (and skill speed) butchers will hit before the next tick allowing you to follow up with flash and stack the deck in your favor of keeping you at the top. Also try opening with overpower to see how it sets you. That has the most effect in a single use instance for war because flash is boosted by the gld passive and war counterpart to that is overpower so while it will delay you from getting to butcher's it will rise you on the enmity chart much higher then the dmg from heavy swing.