Hey, roughly 15 years ago I had a SG done at Kings. I started the process by going to my GP and simply asking for a referral to a max fac specialist at Kings for ‘jaw problems ‘ I kept this purposely vague but mentioned things like sleep apnea and pain in jaw joint. At the Kings consultation I was honest and said primarily my motivation was aesthetic. The surgeon was really sympathetic and said he would try and help. He talked with the orthodontist to work out a plan for jaw surgery but she was not as accommodating (I assume I didn’t meet the criteria you mentioned). As a result the surgeon offered to do a sliding genioplasty.
The procedure went well and he did a pretty large movement st around 11mm forward and 5mm down. Obviously my jaw is not perfect and I still would like more projection at the gonial angle but it has made a huge difference to the way I look and my confidence.
Anyway I hope this helps feel free to message me.