Egyptian passport holder, living in Kuwait, on Kuwait residence, also Bosnian passport holder.
Traveling to Schengen, going out of Kuwait with Egyptian because of Kuwait residence in it, entering Schengen with different passport.
Here is your procedure:
At the check-in counter at Kuwait - show your Bosnian passport (otherwise, you won't get your boarding pass).
At Kuwait immigration, show your Egyptian passport - otherwise you won't get authorization to leave; this is the passport that will be stamped with your exit out of Kuwait.
At the departure gate, show the Egyptian passport.
At arrival at your destination, show your Bosnian passport - because this is your authorization to enter the country. They (may or may not) stamp this passport - this depends on the individual procedures at the port of entry.
When you are on your flight back to Kuwait, show the Egyptian passport with the Kuwait resident visa stamped to get your boarding pass.
At the immigration/exit point - show your Bosnian passport if it is required to be stamped with any exit stamps (again, if required).
At arrival in Kuwait, show the Egyptian passport to immigration for your entry stamp.