There is no such thing as a "layover". There are "connections" and "stopovers". For IATA carriers on international routes, a "connection" is <24 hours or the next flight if that is more than 24 hours later. Anything greater is a "stopover"

Any carrier can book whatever you want on its own metal as well as codeshares, partners, and simply other carriers with which it has an interline agreement. But, this does not necessarily mean online and it does not mean that every front line agent at a call center will know how to do what you want.

If you use ITA and write down the following details:

From -- To -- Carrier -- Date -- Flight -- Fare Basis

and then read those back to the agent in order, that will help a great deal.