Will it really?
"a large secondary display will deplete your battery"
Why would it do that? Instead of powering a high resolution display directly from the phone, all it has to do is send some data to a separately powered, almost certainly lower resolution (most likely still 1080p) display. Unless you're trying to run both screens displaying different things at the same time (is that even possible?), using a monitor with the phone simply acting as a very small PC tower will give huge savings in battery, not deplete it more.