First, are you sure you need a soil movement permit (we call them grading permits where I live)? Typically, those permits only apply when large volumes of soil are moved around and they are intended to A.) make certain that you aren't going to flood your neighbors property; B) aren't going to affect roadside drainage; and C) make certain that your haul-route isn't going to damage the roads.
However, if the building department says that you'll need a permit, you really have no choice
I think you should determine the minimum amount of soil removal is required and then figure out if that's enough to meet your preference.
Start by identifying the existing floor elevation that you want your new floor to match. From there, start working down (add the finished floor height, plus the subfloor, plus the joist depth) to the bottom of the new joists. From there, add on the minimum clearance (required by your local building code) from the bottom of the joists to the ground. With that information, you should be able to figure out how much soil you minimally are required to remove.
If you want to remove more than that for ease of access, just change your height in the volume calculation.
For my house, some of the crawlspace areas are high enough for me to sit upright and get work done, other areas are closer to 20 inches and it is a much tighter fit. I think it comes down to preference and how often you plan to access the crawlspace in the future. Oh, and cost!