Alterac Valley of Olde

You get both mounts for doing the achievement, no need to redo on the opposite faction.

Not sure if anyone mentioned this, this helped me on my progression for the 200 badges A LOT.
Outside of grinding the repeatable quests, make sure you do both side's horde and alliance (turn on mercenary mode located in new dalaran sewers).

The initial quests of capping gy, taking a tower and several other ones that reward 10 badges can be done once on both factions. This will give you a huge boost and lessen the amount you can have to grind via repeatable.
[quest=57302] Alterac Valley Graveyard - Cap a graveyard [10 timewalking badge]
Armor Scraps Armor Scraps - 20 armor scraps looted off corpses of opposing faction members [10 timewalking badge for the first time, 1 timewalking badge repeatable]
Capture a Mine Capture a Mine - capture either mine for your side [10 timewalking badge]
Speak with our Quartermaster Speak with the quartermaster [10 timewalking badge]
Korrak the Bloodrager Korrak the Bloodrager - spawns at Snowfall Gy at the beginning of the game [10 timewalking badge]
Towers and Bunkers Towers and Bunkers - capping a tower or bunker [10 timewalking badge]

These quests can probably be redone as both alliance and horde, however I didn't have a chance to test it out.
Master Ryson's All Seeing Eye Master Ryson's All Seeing Eye [10 timewalking badge]

When you do these two repeatables, I am not sure if it was not respawning or a lot of competition to loot them, but I could not find any when I had mercenary mode on.
Coldtooth Supplies Repeatable Coldtooth Supplies - looted from the Coldtooth mines [2 timewalking badge repeatable]
[quest=57322] Repeatable Irondeep Supplies - looted from the Irondeep mines [2 timewalking badge repeatable]

This was an option a lot of people grinded but I personally didn't do this quest at all.
Empty Stables Repeatable Ram quest [5 timewalking badge repeatable]

A lot of options and choices to grind your 200 timewalking badges. I hope some of this information helps you in grinding the mount.

For clarification purposes, in case it wasn't clear: you want to queue for Korrak's Revenge.