According to the information you provided, the tax authorities took some measures, including mediation, issuing a notice of correction within a time limit and imposing a fine, but the intermediary company still refused to invoice. In this case, the tax authorities can take the following measures to deal with it:

Strengthen the supervision and inspection of intermediary companies. The tax authorities can strengthen the supervision of the intermediary company, regularly check its tax payment, to ensure that it complies with the provisions of the tax law, and issue invoices in accordance with the provisions.

Cooperate with other relevant departments for joint law enforcement. The tax authorities can jointly enforce the law with other relevant departments, such as industry and commerce, public security and other departments, and investigate and deal with the illegal acts of the intermediary company.

Solve disputes through judicial channels. If the intermediary company still refuses to issue invoices, you can consider solving the dispute through judicial channels. You can apply to the tax authorities for administrative reconsideration or bring a lawsuit to the court, asking the intermediary company to issue an invoice and compensate you for your losses.

It should be noted that the tax authorities generally adopt a combination of mediation and punishment to deal with such problems. If the intermediary company still refuses to cooperate, the tax authorities can take more severe measures, such as suspending business and revoking business license. At the same time, you can also protect your rights and interests through legal channels.