The usual flow sequence is: pressure tank > acid neutralizer > AIO iron filter > softener > faucets.
You said the faucets sputter when they are turned on. Is this every time water is drawn, only first time in the morning every day, or only first in the morning on certain days that may relate to when a water treatment device has regenerated or backwashed during the night?
When you refer to bypassing
the system, is that bypassing all 3 treatment devices or only the AIO filter?
If there is too much air added by the AIO filter, I would anticipate excess air will become trapped within the softener. For air from the AIO to pass through the softener, the softener's resin tank would need to become substantially filled with air.
The only part where air could be introduced into the plumbing is the air eductor.
Air can be present in the water if the softener's air-check valve is not fully closing during the brine draw stage of regeneration.
Assuming the softener's control valve is also a Fleck, the air-check valve will be located at the bottom of the brine tank. The air-check is a floating ball that will seal off the brine pickup port when the brine level becomes too low so the ball will no longer float. Occaisionally, dirt or debris (ex: string from a salt bag) can become lodged between the air-check ball and the pickup port, thereby preventing an airtight seal.