There are too many conditions and too many contradictions in this narrative. 

One item not discussed is selling this rifle or passing the rifle on when you quit shooting. The confusion of caliber name, added to chamber and bore size will hopefully cause questions. Few average shooters picking up the rifle will see or understand the ".32-20 CPA" designation. 

The average shooter buying this rifle in the far future will look for .32-20 loading data and proceed. Loading the wrong charge can wreck the rifle with possible injury or death to the shooter. Loading the wrong diameter bullets will cause misses or excessive pressure. Loading the wrong charge can blow up the rifle and then create civil liabilities. 

Thoughts:

• The .32-40 is established for accuracy with fixed and breech-seated bullets.

• The .32-20 is established for accuracy with light bullets at a .312 bore diameter.

• The .32-20 CPA is a specialty caliber designed for the specialty shooter.

With these thoughts in mind, chamber for .32-40 (.321 bore) if you are using fixed ammunition; chamber for .32-20 (.312 bore) with special throating for the heavier, longer bullets if you wish a true .32-20; and chamber for .32-20 CPA (.321 bore) for breech-seating.

One could always paper patch a 308 bullet up to 322 and breach seat that to fire form the cases.
After that there would be no issues.
I have done it with. 32-20 brass to for 310 cadet with no issues.
A half patch making a bore riding nosed type projectile works well lube and not driven to fast.
That or a few grains of fast powder and grits compacted in it to fire form the 32-20 brass; up to chamber specs.