There are two possible approaches for achieving the desired result - to edit the control template of the RadListBox control or to create an attached behavior in which you can modify the composing elements and add the CheckBox programmatically.
Please note that in both cases, the required modifications may be different based on the theme you're using.
For your convenience, I've prepared a small sample project which demonstrates how to achieve this via an attached behavior for the Office Black theme. If you're using another theme and this approach does not work for you, please let me know which theme(s) you use and I will gladly try to update the solution to work for your particular scenario.
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