Does a British citizen have a right to stand as a candidate at a local European election in any EU country?

Yes.

This is one of the rights of EU citizens established by the Treaty on the Functioning of the European Union. This treaty provides for citizens to be "directly represented at Union level in the European Parliament", and "to participate in the democratic life of the Union" (Treaty on the European Union, Title II, Article 10). Specifically, the following rights are afforded: Political rights

• Voting in European elections: a right to vote and stand in elections to the European Parliament, in any EU member state (Article 22).
• Voting in municipal elections: a right to vote and stand in local elections in an EU state other than their own, under the same conditions as the nationals of that state (Article 22).

Note this does not include the right to stand in national elections. This can only be done in your own country.