Over stay problem

A few things first:

1) Wait for a response before sending private messages
2) Your comment on " when someone wants to be legal here system only wants them to be stay illegal I think that's why people overstay in Bahrain " actually is NOT correct and does NOT apply to your situation.  If  your sister wanted to be legal, she should have left or gotten the visa extended - the system allows for many ways to be legal. Don't blame the system for your own mistake.  Your sister overstayed illegally and at the end of the day, it is her fault, not the system
3) The problem between VFS Global and NPRA is not a system problem either.  The problem is that your sponsor is not acknowledging the overstay to avoid future issues - this I also understand as they sponsor many people and can't be expected to track who has left or who is overstaying (they don't have access to immigration systems).  Again, blaming VFS and calling them a fraudster is yet again not taking responsibility for your actions

You have two options:

1) Engage a lawyer to resolve this issue if VFS is not cooperating
2) Utilize a strong PRO to get this through immigration without the paper - which is entirely possible. 

At the end of the day, your sister will have issues in getting visas for Bahrain in the future.  I understand you are upset and frustrated but I hope you realize that this issue is ENTIRELY your fault.  Everything else is adding onto it but it is not immigration or VFS who are to blame for you being in this mess to begin with.