This is your best and maybe only chance to amend the soil via tilling anything in. I would use much more than an inch, though some is better than none. Sounds like a big job. A little more information might help.
Is there irrigation in the existing lawn?
How many square feet, approximately, will you be replanting?
You pretty much have to respect and work off existing elevations of things you cannot change such as top of foundation wall, street or curb elevation, sidewalks, driveway, bottom edge of existing fences, etc. Determining what you can fit within those figures will give you an idea of what you can do without removing any existing soil - which may not be a bad idea. You will need to have good surface drainage. If there are low spots or changes in slope you cannot change, now is the time to provide for the removal of water from those areas via drain tile or French drains. If you proceed without providing a way for excess water to leave the lawn it will always haunt you, especially with heavy soils.
Straight KBG is going to be tough to establish and maintain. It can be done. Most people are not patient enough to wait as long as it takes. It sounds like you are generally on the right track. Be careful pricing it.