How Can I Move One Tooth into Position?

We would "reset" the aligners with new impressions.  This is called a refinement.  This should get your canine down where it needs to be.

Invisalign is a great product that is very technique sensitive and does have its limitations.  Canines are the largest rooted teeth in the mouth and therefore, the most difficult to move. I would continue with the Invisalign and see what happens.  Don't be surprised if it is still not moved into the exact position that you would like.  At that point you could place a veneer on that canine to correct it completely.  I noticed that your canine has a long, pointed appearance so by finishing the case with a veneer you could change the shape of that tooth and correct those issues.  Finally, whiten your teeth before doing the veneer and I think that you will have a result that you can be happy with.   

Teeth move over time when force is applied to them.  In order to put force on your canine tooth, the other teeth have to act as anchors.  They resist the force that is going in the opposite direction to your canine.  Because the canine is such a strong tooth, you need to use your other teeth as anchors to be sure that the force moves the canine.  You mentioned that you want the solution to be "virtually invisible".  Invisalign is the best "invisible" approach, but may take longer than braces because of the way the forces are applied to your tooth.  Don't get discouraged, trust in the process that your orthodontist is recommending for you.  Ask questions, communicate your concerns.

Good luck!

Since the canine has such a long root, it can take more time to move it into place.  You mentioned that you want it to move 3mm.  There is not a QUICK way to do that, and sometimes you can't do everything with Invisalign.  Your orthodontist probably has a lot of experience with cases just like yours and wants to recommend the treatment that will get the best results.  You'll just have to weight the options your orthodonitst gives you, and decide based on what is most important to you.  If you want the best long term result, it may be braces in this case.

It sounds like your orthodontist is right to me.  The braces may be able to rotate it faster.  Hang in there with it, either way.  You obviously wanted it straight or you wouldn't have started it.  Sometimes things are tougher than we think they will be.  Just keep remembering that you'll be so happy when it's right.  Best wishes!