Conector MySQL EF6

você deve manter ambas as seções de configuração, assim
  <system.data>
    <DbProviderFactories>
      <remove invariant="System.Data.SQLite" />    <add name="SQLite Data Provider" description=".Net Framework Data Provider for SQLite" invariant="System.Data.SQLite" type="System.Data.SQLite.SQLiteFactory, System.Data.SQLite" />
      <remove invariant="MySql.Data.MySqlClient" /><add name="MySQL" description="ADO.Net driver for MySQL" invariant="MySql.Data.MySqlClient" type="MySql.Data.MySqlClient.MySqlClientFactory, MySql.Data"/>
    </DbProviderFactories>
  </system.data>

  <entityFramework>
    <providers>
      <provider invariantName="System.Data.SQLite"     type="System.Data.SQLite.SQLiteProviderServices, System.Data.SQLite.Linq, Culture=neutral, PublicKeyToken=db937bc2d44ff139" />
      <provider invariantName="MySql.Data.MySqlClient" type="MySql.Data.MySqlClient.MySqlProviderServices, MySql.Data.Entity.EF6"></provider>      
    </providers>
  </entityFramework>