I would go for enchanting, and take the +6 all stats on your rings and then just go for the badge reward: Bloodlust broach. Imo this alchemy trinket isn't worth it.
You are forgetting to mention that as an Alchemist (Master Elixirs proc extras (including flasks) without extra mats) you can make your own Major Agility, Assasin Flasks, heal pots, etc. Otherwise you will be paying the high prices on AH before each raid. I would also mention an Alchemist can make meta jem diamonds and primal might.
Consider all the pros and cons of any profession. I highly doubt an enchanter is making a tenth of what I make off selling pots, elixirs and flasks.