Renter should sign their own name. But the situation

has already happened. It's not cost effective to fight them in court. In this case, tenant can show that you failed to respond to their contact/request, the judge will think it's your fault. So, the chance for you to win is very small. You should take this opportunity to add redundant communication channels, such as providing a second phone line of contact, or call someone who can relay the message for you. You also need to educate tenant to try best to contact you. If they don't contact you first and get your approval, you won't pay the bill.