Illegal Substitution or Encroachment
 

A has ball on the B 35 yard line. A is lining up and line drops down into their stance. B2 realizes he is the 12th player and coach is at the 50 screaming at him. Rather than sprinting directly to the sideline, take the straightest and shortest route, he run between the tight end and the wide out, through the back field of the offense and is on the 45 yard line, still 4 yards on the field when the ball is snapped.
Rule book says that players can cross enter and exit the game crossing their opponents side of the field.
Rule book also says, once ball is set and whistle is blown ready for play, no part of the offense or defense will cross into the neutral zone.

So has this 12th player committed encroachment, dead ball foul?
Or Live Ball foul for illegal substitution? or something else?

Not sure what rules your referencing, but NFHS 3-7-2 requires, "a player, replaced player or a substitute is required to leave the field at the side on which his team box is located and go directly to his team box." The proscribed penalty is Illegal substitution as a Dead Ball foul.

Sounds like encroachment to me. He broke the plane of the neutral zone. His reason is irrelevant.