Trademarks and copyrights are different rights. If someone else has registered a trademark, you can also register a copyright, but the premise is that the trademark was designed by you. If not, you may be sued. If the copyright registration is successful, then you can charge the other party's trademark for the copyright, depending on the agreement between you.

After successful registration, the other party’s trademark rights cannot raise copyright objections. Same as above, they are not rights of the same nature and there will be no conflict between them.