Since we don't know each other, it's safer to ask.
At 12kmi, your standard head bearings are in need of service and possible replacement. Were they inspected when the forks were serviced?
Have you set the suspension up properly for your weight and riding style?
The short answer is yes, lowering the front end of your bike will steepen the rake angle and shrink the trail value which will make the bike steer quicker. 3mm is not a significant change.
Conversely, if your rear shock is collapsed, overworked, or otherwise insufficient for the task(as most oem are), the rear of your bike will be sitting lower than intended. This will cause the opposite of the effect you're looking for. Compensating for this by lowering the front is a band aid fix and can cause other issues like ground clearance and wallowing of the chassis.
I'm not trying to talk you out of modifying or improving your bike... I've got more money and time and extra parts in my VFR than anyone really should... I'm trying to help you make it the best it can be...