Hi,
I found non linear start-up issue at LP3878-ADJ as attached. It can be seen with Cbypass=0.068uF though it's solved by increasing Cbypass to 0.15uF.
Can you give your comment why this behavior can be seen by using small Cbypass?
Please let me know if you need more information.
LP3878-ADJ Non linear start-up issue.pdf
Best Regards,
Sonoki / Japan Disty