I think this should be the idea: Light Cavalry gets light and fast horses, while the Heavy Cavalry get stronger but slower horses. This was the whole case back then, and continued to be even in the Crimean War. British civilians liked the Light Cavalry more because they were galloping with their curved swords out in the sun, while the Heavy Cavalry was not liked just being it was cautious, disciplined, and serious. Heavy Cavalry had strong but slow horses, while Light Cavalry were some quick riders. In the Charge of the Light Brigade, the Light cavalry lost to the Russians, and should have been counted as a serious defeat, but it was not by British civilians.
Give fast horses to Light Cavalry, and slow horses to Heavy Cavalry, and give the heavies some serious swords, and we'll be alright as far as Balance goes.