MBWC using two twin breakers with middle handles tied
This is almost right

While circuit 2 tripping won't cause the breaker for 3 to trip (nor will it happen the other way around), it won't need to -- since there are only 120V loads present, a fault on circuit 2 will be de-energized by the breaker for circuit 2 tripping, and likewise for circuit 3.  (You'd need a common trip breaker to have both circuits de-energized at once should the breaker trip.)

But, you need a proper handle tie for this

However, a screw does not a handle tie make.  All current production breaker lines have a proper handle tie available for them, although you will likely need to go to an electrical supply house to obtain one, or order the correct one for your breakers online provided you know what make and model breakers you have.

In your case, by the way, your Eaton BD1515s need an Eaton THS1 handle tie.