lost passport traveling on ETD overstayed visa can I just pay fine at airport and fly

My daughter lost her passport and has overstayed about 6 weeks. She has a flight booked and an emergency travel document issued by British consul. She has had difficulties securing interview with PSB ie has gone and queued all day and her number hasn't been reached. she says all her documents have been seen and acknowledged but there wasn't enough time to conduct interview that day. Now its looking like seven working days for issuing it would jeopardise her flight .
Question is could she just turn up for her flight pay the fine and fly out of China back to the UK she has no desire to return in the future.