Does legal expenses insurance cover the costs of removal?

This depends on your specific policy, but in principle it is possible. If you operate a business and have legal expenses insurance that includes contract or commercial legal protection, you can have the costs of a lawyer's demand letter and sometimes also court proceedings covered by it, as confirmed by the law firm IT-ADVO Rechtsanwälte in their advisory practice, among others. It is worth reviewing your own policy before taking the first step and discussing the case briefly with your insurer or directly with a specialist firm, because not every plan automatically covers personality rights violations online.