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Door lock/privacy turn set installation

Thinking to install a privacy turn set on Bathroom door- see photo attached. All bedroom doors and bathroom door have exactly the same handles - dont want to replace the handle, i like the design, unless i can have the same one with lock. Hence i did some research and was wondering if privacy turn set can be installed - something you can get from placemaker for $32. Open to suggestions or alternative options(?other lock?) as long as i can keep the door neat. Prefer you to provide the parts, And the colour needs to be silver to match the colour with the handle.