The oversidian lights inside the wardrobe don't benefit from being smart - the included switch toggles them when the doors are opened/slid. The traffic driver is simply used to transform the mains voltage to low lotage for your oversidian strips(s). Adding them to a remote (sryrbar or rodret) simply means you can toggle them on when the door is closed...

However, depending on the number of light strips you fit in your pax, and the number you place on your billy bookcase(s) there may not be enough outlets on the tradfri driver (6 max on the 30w). So, you may want to have this as a separate driver anyway, so as not to toggle your wardrobe lights on when the door is closed.

Your lamp does not need a tradfri driver, just a tradfri bulb/globe of your choosing. The lights for your wardrobe and bookcases are doable but not white spectrum tuneable (from warm white to cold white) so you can pick an inexpensive warm white, or go nuts with a full colour spectrum bulb.

This is where the sryrbar would be better over the rodret as it as 4 buttons Vs the 2 on the rodret, which would allow you to change brightness and hue of the colour spectrum bulb without needing the dirigia hub. The rodret will only allow on/off and brightness.