You can as I stated install mini split ducting into the soffit's, because they are flex duct. You just have to be able to get a fish stick in there to pull it along the soffit, if there is enough open space up there. If not, that means opening up sections along the length, removing any insulation materials that are packed up there, or manually route the flex duct through the soffit to where you want it. Keep in mind that depending on the size of the space, dictates how many flex ducts will be needed to supply air.
As for placing the unit behind the screen, there are ones made for being placed out of the way, with a return that would be under the screen. Otherwise, you are still going to have to have to locate the inside unit in say the utility room, with the back of it against the wall where the screen is, or near where you can run a duct system, for the returns, but also allow routing of the supply on both sidewalls for the supply vents into the room.
Engineers call this placing the cart before the horse, and not thinking about the consequences, until everything comes to a screeching halt, which in turns creates a disaster. Also another reason why you see separate rooms with server room cooling and temperature monitoring systems in those private theaters that were properly thought out from the beginning, not just thought of in a person's head.
Unless you have everything on paper, have worked with environmental engineers to calculate heat load from the equipment, people sitting in there, when you close the door, that is designed to not only seal off noise, but any air exchange too and from the space, you start creating dangerous situations, that then have to be addressed by either retrofitting the environmental systems into the space after the fact, which means yes tearing into the construction, or doing a complete demo of the room, and redoing from square one, because the room turned from being what a person thought would be perfect, but in turn has created a disaster, that just looks pretty, but in reality is not safe in quite a few aspects.