Y'all are over-thinking. Such a player must sit out a 'down.' The ball has to be put in play and subsequently become dead. Forget the RFP period. After he has left the field, and the ball is declared ready for play, if you don't have a snap/free kick and a subsequent dead-ball, you don't have a down. Until you have a down, he can't come back in.
An interuppted RFP period before a snap or free kick is not a down - it is something that precedes a down. You can consider it an element of a down, if you want, but unless you have all of the elements of a down, i.e., RFP period, snap/free kick, and dead-ball, it isn't a down.
However, the case of a period expiring before the next down has not been addressed.