Is a 3mm nose tip lift going to make much of a difference? (Photo)

In my universe, I don't know of any operation other than a real revision that can raise the tip of your nose, even "just" 2-3mm, which actually is a substantial change.And I know you didn't post a full set of face-at-rest photos, but from these two pictures, it seems that your tip could use a more substantial elevation, more like the revision rhinoplasty patient in the short video that I posted with this answer.If you have an operation with "almost immediate results-meaning no swelling, bruising or even a bandage," it can't change the position of your tip, and it would make any future surgery more difficult and less predictable.I'd love for you to see some excellent professionally-designed morphs of what could possibly be done with your nose. Morphs could also help you identify better what's bothering you, from any angle, and help you set a goal for the rhinoplasty that's accurate for your tastes. Profile and three-quarter views would be particularly important in morphing your nose. (Side note: in my opinion, morphs should always be done by the surgeon, or he should direct an assistant as she makes the morphs. Morphs should be made with a constant eye to what actually *can* be done in surgery, for that particular nose, and only the surgeon has that information and judgment. Another aside: I also recommend that you not show morphs made by one surgeon to a different surgeon. If a surgeon makes his *own* morphs, you get to see whether he understands the important issues of your nose, and how your nose should be changed. But if you show the surgeon someone else's morphs, and he says "Of course I can make your nose look like that," then you don't really know if he fully understands, or even recognizes, the important changes shown in those morphs. And you don't know if he can make those changes during surgery.)
Remember that rhinoplasty is an exquisitely difficult operation to get right, and you should only have surgery if you are able to make yourself very confident in your surgeon's skills. The changes that your nose needs require advanced techniques, and skill that most plastic surgeons don't possess with expertise. For most noses, it's much better to not have surgery than to have inexpertly-performed surgery. Read my essay on how to stay out of trouble while selecting a rhinoplasty surgeon. And for those who have had previous surgery, it also discusses how to tell whether your first surgeon should be performing your revision. And it also discusses how to take photos that are best for online evaluations.
Your nose is also a perfect example of why computer imaging by the operating surgeon is mandatory in rhinoplasty. You need to know exactly what the surgeon is planning to accomplish -- what features he thinks he can change, and by how much he thinks he can change them. I always email morphs to my online rhinoplasty consultations. When you see the surgeon's goals in the morphs, you'll know whether he has an eye for an attractive nose, and whether he shares your opinion of what constitutes an attractive nose. You'll also know whether the changes he proposes are large enough to be meaningful to you, and whether he understands your wishes enough to address all of your priorities. How would you ever get that figured out without the morphs?Whew! Sorry to throw water on this idea, but you asked!!

I have found that small elevations of the nasal can be done simply in the office with internal stitches only and no need for a nasal splint.  Oral sedatives help.  In your situation that may be a good solution as it seems that you are  otherwise happy with the nose.

Revision rhinoplasty requires exceptional skill on behalf of the surgeon and great communication with the patient about realistic goals and expectations.
Common cosmetic concerns include a dorsal hump, wide nose, sunken nose, irregularities, and asymmetries. Functional concerns include difficulty breathing through the nose, which can and should be addressed at the same time as your rhinoplasty.
In a revision rhinoplasty, cartilage from your septum may have been depleted and therefore often an alternative source of cartilage is needed, such as from the ear. This is just one of the important differences between primary and revision rhinoplasty that need to be discussed in detail before you decide to proceed with surgery. 
In your situation, a profile view would be helpful to evaluate your tip position. Most likely, a minimal procedure as you described would give you minimal results. If you are looking to have the tip elevated in a meaningful way with long term results, then consider a revision rhinoplasty.
Make sure to seek out the care of a board certified surgeon with extensive experience in revision rhinoplasty. Find someone you connect with, who will listen to your concerns and provide you with a framework for what is possible.  
Gary Linkov, MD
Rhinoplasty Specialist
Manhattan

Revision rhinoplasty is more difficult than a primary rhinoplasty due to the fact there has been previous alterations to the normal anatomy, and scar tissue formation. In addition, a full set of facial photographs are required to make a determination about how best to proceed. There appears to be an active depressor septi ligament which needs to be released, and a hanging columella which needs to be reduced.