any media you place in the filter will eventually become bio media.
the difference is how much surface area you v=create to allow beneficial bacteria to attach to.
the sponge does more than just filter out the sediment. it also has surface for the bacteria to set up house.
the bio media(ceramic rings etc) basically occupy the same space but have more surface area for the bacteria to establish.
adding more of this media will not speed up the cycle..but when the cycle gets established will help make for a good base.
sponges need to be rinsed out thus you may loose some bacteria the rings may get slightly rinsed from time to time.
activated carbon is useful at helping to remove pollutant..I see no detriment to using it in the filter to help..but it can get costly and is not considered necessary in a tank that get regular water changes.
not to be jerk but you asked a yes or no type question. "will taking out the carbon and adding another bio-ring insert speed up cycling". The answer in NO. The only thing that will do is add more surface area for bacteria to create colonies. No other way to say it.