"When either the Northern Horse or Byron are about to Rout, add the Lifeguard's Unit to that Formation. That is, the Formation does not Rout, but only has one Unit block."
So - when either of these Formations have lost their last stick, it gets the stick from Rupert's Formation. If and when that stick is removed on a subsequent turn, then the Formation would Rout normally.
"Remove this card from play if and when that happens. This does not constitute a Rout."
When the Lifeguard's ability is used, the Lifeguard card is removed from play. Removing the Lifeguard from play does not constitute the Lifeguard Routing, and so would not result in any exchange of Morale Cubes.