What must I do regarding my remorse for falsely claiming that an incident I was involved in was a work-related injury, which led the company to cover the costs of my treatment?

You must inform the company management of the truth and offer to repay the treatment costs or have them deducted from your salary. If they forgive you, the obligation is dropped. Otherwise, your conscience will not be clear until you repay the expenses or they explicitly permit you to keep the money. Seek forgiveness from your Lord for the lying and injustice.