I have used a .257 ackley for many years.I right now have my Nosler loading manual open to the .250 sav cartridge drawing.The chamber cast idea is not bad.Who knows what you have!!
A regular .250 sav headspace gage is not likely to work,it has to do with where the datum is
By design,the AI headspaces a standard round at the junction of the neck and shoulder of a standard round.On the drawing,this is called out as a .289 dia.Upon fireforming,case taper will be reduced .The .414 dia will become larger.The shoulder angle will become steeper.That original headspace "ring " of .289 will not be blown forward.
It is inportant when fireforming that you don't crush this back by forcing a claw extracor to snap over it.feed up from the magazine.
If you handload,you are probably concious of the point where a seated bullet may contact the rifling.I use this to my advantage.The old 117 gr roundnose bullets are perfect for this,but any 117-120 hunting bullet will work.I long seat them so they contact the rifling,and then maybe .015 longer.I want the bullet jamming into the rifling to hold the case tight against the bolt face.
Pick a moderate load in the quicker spectrum of recommended powders.Do not subload.If you were using a 115 ballistic tip,I might use 29n gr of 4895 or 28 gr 3031,both IMR powder,as shown in the 5th edition Nosler book.
At firing,the case volume will increase,and the bullet is seated longer than spec.Both will reduce chamber peak pressure.
Now,it is unlikely these will match your sight in.They may be very accurate.
You sure can practice your position shooting for group size.A nice group shot sitting does not have to be centered on the point of aim to be a nice group,or good shooting practice.(No need to twist the knobs)
When you are finished,so long as you do not set the shoulder back when you resize,the cases will last a very long time.Precise headspacing is one advantage of the AI case.Use the bent paper clip trick to check for an internal stretch ring once in a while.I suspect you will lose your cases to neck splitting about reload 8-10.
Sweet cartridge! Oh,check your twist for an idea what bullets will work well.A 1-10 for 115 to 120 grs,or maybe if it is a 1-14 you will use lighter bullets.