Depends on whether or not you care about the image appearing squashed/stretched.
If you don't care, then you can hold the shift key while dragging the corners of the transform tool to unlock the aspect ratio. You can also do this by clicking the chain link icon between the x and y values of the "Zoom" item in the inspector pane, and then altering the numeric values.
If you care about squashing/stretching, you care about cropping, you care about the image not being too small, and you have a fixed screen size, then the answer is no. There's nothing you can do. It's just the geometry of rectangles, not a limitation of the software. You have to compromise in one of these areas. Maybe you crop an equal amount off of each mirrored half, for instance.
But when you say, "take up half the screen," many of us make automatic assumptions about the size and shape of the rectangle you're trying to fill. We assume that you're talking about a television, most likely, or maybe a YouTube video, but almost certainly something with a 16:9 aspect ratio (a width which is 1.7777 of the height). But DaVinci Resolve will let you arbitrarily decide the dimensions of the movie file you're trying to create. So, you could just set your timeline resolution to twice the width of your source media.
To do this, in the Project Settings -> Master Settings, set the "Timeline Resolution" dropdown to "Custom" and enter numeric values in the boxes below. The left field is width, so you'd enter double the source media's width in that field, and keep the other field the same as the original. If you set it up right from the beginning, when you go to the Deliver tab to export, you shouldn't have to touch the "Resolution" dropdown menu. If you do find yourself reaching for that menu, chances are you've made a mistake in the Project Settings.
Of course, If you do try to play a double-wide movie on a 16:9 television screen, you're going to run into the same problems you had in the first place. Most likely, the TV will add black bars to the top and bottom, and scale your image down to smaller than you wanted. But it could also stretch, or crop, depending on it's default behavior. However, if your end goal was always a movie file that would play back windowed on a computer, you're golden. Just don't think too much about how that window fits into the rectangular display of whatever device the viewer uses ;-}