Overjet/Overbite - Large Bottom Incisors? (photo)

In regards to your bite correction any type of braces you choose should be able to achieve very similar results. The advantage of the lingual braces is that they are invisible since they are behind your teeth, but they will all move them in a very similar manner. Once your teeth are in the right place their size might not be a concern for you anymore.

  Orthodontic treatment to correct a tooth positions can be done using a variety of treatment methods.  Incognito and Damon are brands of braces.  Incognito is a brand of lingual braces and Damon is a brand of traditional braces.  Any orthodontist should be able to use their preferred treatment method and get a great result.  

 

As far as the bottom teeth.  I suggest you see how your teeth look when they are straight and you might see the proportions of you bottom teeth may look ideal.  You can always remove length but it is difficult to add length.  So take a look and decide that when everything is staightened.

It is hard to get the full perspective from the pictures you have enclosed.  However I can say this; you should get a lot of opinions.  Don't go for the treatment because it is what you want to hear.  There are a lot of simulation software packages (e.g Dolphinimiaging.com and Anatomodel) that can show you the difference of the different approaches using a scan of your head with a picture of your face.  You need arch expansion and leveling.  I would not extract any teeth except 3rd molars.  I think you will get best esthetic result if you do surgery.  Depending on where you are located there are some very talented surgeons.  Current thinking is do the surgery FIRST and then the BRACES SECOND.  This is in sharp contrast to traditional methods.