How is a Constitution changed if the majority of the electorate do not want it changed? If a minority is allowed to determine which groups are allowed full rights of citizenship, that would be an autocracy, not a democracy.
It typically isn't changed; it's clarified through a bill or a court ruling. The Canadian Charter of Rights and Freedoms already guarantees the rights LGBT people are seeking in the majority of cases; it was just being unevenly applied when people thought being gay or transgender was some sort of disease or moral failing rather than an innate and/ or harmless condition. Canadians already agreed that equality in certain matters was important. Applying equality to LGBT people is now a matter of reason, not opinion. The process is still subject to human error, but for the most part minority rights can progress somewhat faster than the time it takes for the majority view of the [voting] populace to change.
That sounds good, but what it means is an elite judiciary determines the future of the nation, which takes us back to an autocracy.