I would frame the ceiling with "hatt" profiles (metal studs). It might be a bit more expensive than wood, but it's quick and the material is straight itself. You seem to get some span, so there will be some reinforcement studs that you attach to the hatt profiles and then fasten to your existing ceiling. A laser can be helpful. There might be information on Gyproc's website and brochures you can have sent home. How much are you planning to lower the ceiling?
Best regards, Putte
Best regards, Putte