It would seem sarada is a sensor/tracker type. She wants naruto to eat something so she can track him. That would explain how she will be able to follow without him realizing. She can just stay out of his tracking range and follow him via whatever she made him eat.
Mitsuki got the digits lol
Naruto already knows she's there. And How will her sensing/tracking have a larger range than Naruto's? Lol
I honestly don't know. She wants him to eat the food for her own personal reason. That she is going to track him is the only logical explanation. Naruto had sage mode active because he already sensed her outside his door. After collecting the food, he is probably going to send them home. In which case he wouldn't need sage mode anymore would he? So she can track him without plot problems.
After finding Shin attacking them I can't imagine Naruto will send them home without supervision. He could make a clone to escort them back.
Salad wants to bring the lunch to him as an excuse to follow, in the instance she is found.