So after some thinking and some frantic research, I have come up with a civ for this particular challenge. So I hereby present to you, Batu Khan of the Golden Hoard!
UA: Khan, Tsar and Shahanshah
Before the industrial era, may win the game if you own a third of the world's cities and three foreign capitals, as long as a third of your empire is

puppeted. Mounted units gain +1

movement and an additional

movement point in open terrain. +5%

gold in puppet cities.
UU: Mingghan
Unique range mounted unit that replaces the knight. +1

movement, though its weaker than the knight(17

\

). Has the great generals I and accuracy I promotions.
UU: Jochid
Unique great general replacement for the Golden Horde. Rather than the great general's abilities, the Jochid grants +20%

strength to friendly units and may be used to subjugate enemy cities once, bringing them under your control as

puppets while retaining 66% of their population.
(Doing so does not consume the unit and requires 2 adjacent units)