MessyLife
option A multiple reasons, mainly being it will strengthen thier bond and trust between each other, Lying doesn't seem like a good option remember they are ninjas not some half baked kids, specially sayuri, I am sure she has her fair share of deducting if someone is lying or not ( while working as a spy for orochimaru previously, she should have been trained enough for it) besides it depends on how yamato sensei goes about it, if he helps then good, Karin's favor has been somewhat repaid if not then shinji can use it as a catapult ( or reason) to become more stronger and influencial in the future,so he can repay karin the favor since The promise is to drop this specific matter but not that he isn't allowed to pursue it when he can and has the power for it without her, this would have gone in a very different way, we might have even had lost kota due to blood lost or some other horrible outcome either way, lying would probably backfire and minus some points with sayuri, otherwise a truthful promise can max her trust for shinji more and she might open up more to shinji and trust her with his life
A.There is no reason to lie or go that far for karin honestly and after talking to Yamato and see what he thinks about this should be the end of Shinji repaying his debt.He did what he could to help her when she saved one of his teammates from at worst dying and at best ending his ninja career and his hopes and dreams at becoming hokage.