Alter not part of Mayor's Cup
Alter is a private, parochial high school, therefore, other than being located in Kettering has no "govermental" tie to the City of Kettering. It this world of PC and separation of church & state, some may take offense that a Catholic institution would represent the city.
As for tournament games counting for the SWOHSHL, I believe the fact that the games are scheduled by mutual agreement with SWOHSHL teams, played under league rules, with league referees ... I am not sure what else would be done to make them count. Everyboddy played everbody and the winner was decided based upon record. Tie breaker would have been head to head, then goal differential. Teams did not get points for winning periods. It was like real hockey games, imagine that.