I have not found any way to specify the Postgres schema that mattermost should use and saw that the "public" schema was in the sql migration files.

Using the public schema is not a recommended practice with postgres : https://wiki.postgresql.org/wiki/Database_Schema_Recommendations_for_an_Application

See first section stating :
"The recommendation is to create a single database with multiple named schemas [...]. Additionally, it is recommended to remove the public schema."