Well best Han China will be the Songs
.
Roman are disciplane. China can use rocketries and fire carts to blow at the Romans, first time for Romans to see such fire work! Therefore, away they flee! And Roman cavalries charge and the Hans will charge with Cavalries too (Hans infantries has ordinary armour you can say, but lot other higher rank soldiers got iron breastplates that covers the whole body, and the Romans only got legion armours or even leather. China has an big army in the Medieval times too.
I would say Romans will lose
. Just cuz they got nice formations does not mean they got better weapons. We all never seen Romans fighting Han China. So it must be called a match.
But for saying, I vote for China.