I used to work with a guy whose mother was a civilian employee at Tinker. She knew how to "work the system". The last time I saw him, she hadn't worked a day in four years and never missed a paycheck.
From the time frame you mentioned. She won't get any unemployment benefits as she has not worked enough with the company to qualify. Second, a simple letter stating she was not terminated and detailing the facts of the events would be enough to get the claim denied. Third, Whoever is doing your candidate screening sucks.
As far as not costing the company any more money. Immediately not fighting the claim won't cost the company money. However, the next time the company's experience modifier is updated. Then the company will see the cost. The more claims you have the higher that goes. The lower the number of claims the lower it can go. Granted this is all minor increases or decreases. However, It is a factor that goes into calculating state unemployment rates.
It also sounds like your HR department and supervisors have a problem keeping their mouths shut. Outside of HR and the employee's immediate supervisor. No one should know if a former employee files for unemployment unless that employee discloses it with their prior coworkers.