Please note that LINQ Reporting Engine capabilities are restricted by MS Word capabilities. MS Word does not provide an out-of-the-box way to use icons for chart categories and place categories the way you describe.
That said, there is a way to generate a chart as per your screenshot given that categories are the same for all reports generated from a single template.
If categories are always the same, you can just use picture elements and put them over a chart element. The same can be done for category names - you can use textboxes the same way. To have some extra whitespaces between bars of chart series, you can add a series with all zero values, so it does not appear in a chart but just adds some whitespaces, which can be used for textboxes with category names.
Please check Template.docx (53.7 KB). It shows a result of the mentioned steps.