These two projectors use very different technology and there are pros and cons to each. The JVC uses a version of LCoS, or liquid crystal on silicon, in a three chip array that converges together to form the full RGB image. The main advantages are a drastic increase in native contrast and an image that is more similar to what you would get on a conventional television or monitor. The disadvantages are you are in convergence and the amount of light you can get out of the technology. The BenQ uses DLP technology, which uses millions of tiny mirrors to point the light to the right place on the screen. It can do this very fast, so it can handle a lot of different frames per second, allowing for a color wheel to be used to flash the different colors up on the screen in rapid succession to achieve the full RGB spectrum. The main advantage is there is no convergence required, so it can have an impressively sharp image, and the light path is simplified and allows for much more light to be passed through the system. Also due to how fast the DLP mirrors can move, you can get impressive motion interpolation or processing, making sports games look better. The main disadvantages are severely low native contrast, lack of color accuracy and color gamut, and because it flashes the different colors at different times between each frame, some people can see what flashes of rainbow colors if their eye flicks fast enough across the screen. This is referred to as "RBE" or Rainbow Effect. Some people are more sensitive to it than others.
Typically DLP projectors under $10k don't have a lot of the same features as the higher end LCoS, such as a large amount of lens shifting, powered lens shift, focus, and zoom, and a host of other small features. This is probably to keep costs down, as some of the new BenQ models I have seen have more lens shift, and adding power features would be relatively easy, they just don't seem to offer them.
Bottom line, is how each would fit your needs. Since your theater is not "blacked out", you wouldn't get the full benefit of JVC's very low black floor, but that isn't to say that the improved native contrast wouldn't be a huge benefit. If you are used to watching OLED TV's and like the deep rich blacks, then you would probably prefer the JVC over the BenQ. However, if sports is one of your main priorities, then the DLP's extra lumen output and the sharp image with good motion might give the edge to the BenQ. It looks like you have a 16:9 screen, so you don't need to power zoom and shift to go from 16:9 to scope format, so a lot of the little features of the JVC might be lost on you. If you game a lot, it is a mixed bag, as 4k gaming on the JVC will give you the best image quality (being native 4k and able to resolve pixel details better), but the BenQ may have lower latency by about 10-20ms.
Overall, the biggest thing people notice is that DLP's throw a bright and sharp image, while the LCoS's throw a rich, colorful image with great contrast that tends to be closer to "looking through a window" rather than looking at a screen. Most people on this sub-forum would tell you without a doubt to go with the JVC, but there are some holdouts who would say to give the DLP a try. They each have their strengths, and about all I can say is I went the JVC direction, mostly because of features, even though I have a very large screen and would benefit greatly from the brighter DLP.