I may be wrong, but a "class 3 license" is not needed to purchase NFA items. Form 4, tax stamp, and chief LEO signature are needed.
The corporation loophole is that corporations do not require the chief LEO signature, so in jurisdictions where the chief LEO will not sign off, having a corporation make the purchase gets around this.
The downside is that if the corporation is ever dissolved, the weapons must be sold or otherwise disposed of.
As to whether a sub S corp. will work, I don't know.
"Those who live by the sword are probably pretty f***ing good at it."
"Instructions for a successful gunfight: Front Sight, Press Trigger, repeat
as necessary." - B. Braxton