When someone damages your property and refuses to pay, your recourse is to sue them. If you sue the mover, you will have the opportunity to present testimony and evidence that they caused the damage and that you never signed the report; you can also challenge any evidence or testimony they present. For several hundred dollars of damage, you should sue in small claims court  if possible, representing yourself (that will let you avoid legal fees); if several thousand dollars are at stake, hire an attorney to represent you.