Well, it's definitely an interesting idea, although I'm not sure it would actually change all that much. I suspect that a Tohsaka Sakura would act much as her sister did, and the same for a Matou Rin. The situations they're both in in canon are extreme enough that their natural personality differences (which aren't that great to begin with) would pretty much get washed out.
Having said that, their reaction when the situation unfolds might well be different. Rin, for example, might well be more resistant to Zouken initially, and would probably then break hard later on, and thus by the time of FSN would probably be less able to resist him, even passively. As a result, I expect she would end up fighting in the war. Sakura, conversely, seems a bit more prone to blaming herself for things than Rin and a little bit "softer", so she might reach out to her sister a bit more, or at least show less willingness to fight her. However, Sakura also has a lot of determination and mental strength, which could mean she would stick to her goal more.
It's difficult to say how they would change, though, because their personalities in canon are so heavily affected by their childhoods. I suspect a Matou Rin would resemble Matou Sakura more than Tohsaka Rin, though, and the same would apply in reverse for Sakura. Their natures were heavily influenced by their upbringings, and HA demonstrates that Sakura ends up becoming much more Rin-like once Zouken is out of the picture.