It can be very difficult to sue union for inadequate representation. Union is not necessarily even required to hire a lawyer to present or argue your case. You have to read your collective bargaining agreement. May well require legal consutation to fully understand your rights. If the allegation of lying in an investigation is true, you have nothing to pursue.
FMLA issue may be different. The FMLA is intended to protect workers who must take leave and qualify under the statute. Either you gave proper notice and legitimately qualified for FMLA benefits or you did not. The FMLA rules can be tricky so care is required.