If you are just getting into things and budget is a bit of a concern, get the HT2550a and be completely happy with it for several years.
Understand that HDR is something of a myth with front projector (there are plenty of articles as to why) and while 4K HDR content does offer some advantages, they are not as night and day as the original jump from SDTV to HDTV was.
Now, 12' and 4K is not a real thing. It is about how far you are viewing the image (12') and how LARGE the image is. People say 4K is only worthwhile when you are viewing a 60" or larger TV. That's just BS. You sit 24" from your computer monitors. Those monitors may be 24" in diagonal or larger. You really think that jumping to a 4K image when sitting 2' away from a 24" (or larger) monitor doesn't make a difference? Of course it does!
It's about viewing distance AND image size. So, from 12', if you are viewing a 150" image diagonal, you will see a jump in quality between 1080p and 4K. But, at 120" diagonal or smaller, you won't see as much of a difference.
Now, the big jump with 4K is not just the resolution, but the jump in color. This is a good thing as any display which accepts a 4K source may be able to use that additional color depth to improve image performance. That's nice.
The key being that you are very unlikely to have ANY sports viewing which will be in true 4K and certainly none which will likely take advantage of the additional colors afforded to 4K content. It's just to much bandwidth to take up anytime soon. So, your sports viewing is likely to be 1080. Maybe not even 1080p. Sports, for years, has been broadcast at 720p resolution. Broadcast technology is YEARS behind display technology.
NOW - Sports viewing!!!
The reason that people talk about the TK800 because it is bright. It doesn't have the contrast of the 2050a, but it certainly will be brighter. It won't have the color saturation... but it will be brighter.
Do you need that brightness?
NO! Not if you make a pointed effort to control the ambient light within the room you are in. That is, you can't have half a dozen recessed ceiling lights on, especially near the screen, while using a projector for ANY viewing, including sports. But, that doesn't mean you can't have light on. It just means those lights need to be away from the projector. Install, or buy floor lamps, which are directional and that direct light downward near the seating area, and use dark seating and dark colors to prevent reflections from landing on the screen. You can have quite a bit of actual lighting on while still enjoying the game with friends.
I use this as an example. It is 10+ years old now, but still applies. The projector in use is a Panasonic PT-L300u (my first digital projector) which may have been putting out 300-500 lumens on a .8 gain 106" diagonal grey screen. But, the lighting was solid in the room it was in. The room had grey walls and ceiling and a dark green main wall.
http://www.avintegrated.com/lighting.html