Oh boy. I haven't touched either Final Fantasy or Inuyasha for a while but as I can remember, Cloud and Inuyasha would be pretty evenly matched. Honestly this fight could go either way.
Round 1: Either way but I would say it would most likely go to Inuyasha. Inuyasha characters aren't exactly known best for their speed but against a character they can keep up with, they can be monstrous. With the speed equalized, Inuyasha can keep up with Cloud and be able to tag him several different times. Inuyasha's arsenal would help give him the edge in this fight as it can literally prevent Cloud from using his arsenal in the first place. For example the dragon scale tessaiga can absorb an oponents energy when fighting them and against stronger enemies with greater energy, the sword can steal plenty of their energy to weaken them. Literally Inuyasha can weaken Cloud's sword or drain it from its energy preventing him from using his attacks just by clashing blades. I say Inuyasha wins with low-mid difficulty.
Round 2: I would definitely go with Cloud. As I said, Inuyasha characters aren't know best for their speed and I think Cloud's superior speed would be too much for Inu to deal with and it would give him the edge. Now as I can remember, Cloud's got that wicked multiple sword thing that Omi showed me once that's just darn amazing. I doubt Inuyasha is durable enough to tank that or fast enough to dodge it. Inuyasha could counter him with a wind scar or adamant barrage which COULD help him win, but with the speed not equalized, Cloud could simply... get out of the way. If we give Inuyasha some room, then he could win but Cloud would still most likely have the victory in his hands, and in a close encounter, Cloud definitely wins, no questions asked. I say Cloud takes this with mid difficulty in long ranged, and low difficulty in close encounter.