How to use Counters ONLY to construct a School Bell circuit?
This is just crying out for a microcontroller.  Even the simplest dumbest that can drive a crystal (for accurate timing) can do this.  That's certainly how it would be done in the real world.

If you've never used a micro before, there is a lot to learn.  However, you seem to be in some electronic related field of study, so you'll have to learn this sooner or later.  The sooner you do, the more you can have it available as a tool and not contort designs to avoid microcontrollers.

So dig in and get thru it.  One of the PIC 12F from Microchip should be able to do this.  Anything that can drive a crystal will do.  A small 12F is easy to learn on because is has too little RAM and program memory to worry about banking and paging.  Since the input to the logic is just the oscillator and the output is a single digital level that would ultimately drive the bells, this can be easily simulated.  You can write and verify the whole program on the MPLAB simulator.

Find a PIC 12F that can drive a crystal (most of them I think), get some coming, read the datasheet, install the free MPLAB, and start playing with code in the simulator.