Is it considered forgery or unlawful for a believing woman to take the full insurance amount if the laboratory owner recorded the official amount of the analyses on the invoice before the discount, even though she paid him a lesser amount after the discount? And what should she do if that is unlawful?

What the employee did by writing a false invoice is forbidden forgery and prohibited deception. It is not permissible to take more than what was spent on the treatment. You must refer back to the employee to record the correct amount, or return the excess amount to the insurance company. If you are unable to do so, then donate the excess to the poor and needy.

There are two types of insurance: the permissible cooperative Takaful, and the prohibited commercial insurance based on gharar (uncertainty) and qimar (gambling). If your work insurance is commercial, then you are not allowed to benefit from more than what your employer contributed. If the increase added by the employee does not exceed the amount your employer contributed, then you may benefit from it.