I've already seen a Ranma x Naruto fic where tests of Ranma's blood show that he has a precursor version of the sharingan. The way that this author handled it was that Ranma seems to have an unnatural, or at least extremely exceptional, ability to memorize exactly what he sees in a fight, and then study it in his memory until he figures it out. He then has to put himself through the physical training needed to actually use those techniques. In many ways, the Sharingan does the same. It did not let Sasuke copy Lee's moves, but it made it much easier for him to train over the month between the preliminary fights and the main fights to learn a lesser version of them because he had a perfect memory of Lee's movements that he could continuosly compare to his own actions to spot the flaws in his training.
Where the Sharingan rules is in copying jutsu, because for most basic jutsu you are using the exact same energies and muscles arranged in different patterns. Once you have those patterns memorized, and already have the movements committed to muscle memory through the seal drills shown as part of their training, it becomes really easy to add a large number of jutsu to your repertoire. There are plenty of techniques it is useless in copying, though, such as the rasengan, because that is using chakra in a completely different manner than he has been trained, so he would have to go through the entire training procedure to figure it out, just as Naruto did. He would have the advantage of not needing it shown to him more than once, but that would be the limit of that advantage.
A version of Ranma born and raised in this world, with an Uchiha mother and Saotome father, could decide that reliance on the sharingan is one of the greatest weaknesses of his clan. He travels with Genma, learning as many different forms of Taijutsu as he can while mastering the basics of ninjutsu so that he is not restricted to either. To keep certain aspects of Ranma's character intact, I would suggest him either having a mental block regarding genjutsu, or Genma(who never mastered it himself and has horrible memories of Happosai's usage of it) keeps condemning the path as weak, keeping him from being anywhere near as skilled in that area.
Since it is guilt that causes the mangekyo, as shown by (technically a spoiler)
- SPOILER - Show Spoiler
Sasuke managing it without killing Naruto
, Ranma could activate it over any of a number of incidents, such as his feelings of guilt over what happens to Ryouga, despair at Jusenkyo before he learns that he can change back, guilt over hiding from his mother out of fear that he would not live up to her standards, etc. It requires very intense emotion, usually coupled with intent to kill, either on the part of the wielder or on someone else, but there are plenty of canon events from the Ranma verse that could qualify as being emotionally intense and life threatening enough to activate it. It could even happen before the start of the story, when Ranma feels like he abandoned Ukyo or Ryouga, especially if you just don't want to write that scene. The great thing about flashbacks is that you can make them relatively vague references.