As much as I love to promote the thrill-factor of projection for home-theater, I'll be the first to emphasize that projection really only works as it should in a darkened room.
If your basement will get good light during the day... and I see that you have windows positioned on the back wall behind the viewers when facing your screen, then this would "wash out" any attempt to project an image. In fact, having windows on the back wall facing the screen is the *worst* configuration for natural light to compete with a projection screen as the windows cast light directly onto the screen in the same direction of the projector. some fancy screens can help reject off-axis light as from a window in a side-wall perpendicular to the screen, but nothing will be able to reject light coming at the screen on the same axis of the projected image.
If your goal is really a family-room to be used with the lights on, then a large flat-planel TV would really be the way to go. I'd suggest going as absolutely large as possible... at least 70 inches. The "wow" factor will still be good, and 3D will still be very impressive on a screen this large if you're interested in enjoying 3D (which I strongly recommend especially for family enternainment wtih 3D pixar and Disney films).
Now... here's a thought... and this is not unusual for "multi purpose" family rooms like yours in the basement...
Go with the large flat-screen for the non-stop kid viewing and enjoyment during daylight hours.
Then, if your budget permits (even in the future as an add-on), add a drop-down screen to come down in front of the TV and hang a PJ for serious "movie watching" at night. With a screen 100" or bigger, it will be no-contest.
The real point to this last bit is that you're not locked into just "one" solution. Sure, with your budget you have to be realistic and prudent, and given your overall situation I'd steer you to spend $$ on a nice 3D large-screen flatscreen. However, down the road if you get-bit by the home-theater bug and really want to enjoy movies in their fully glory, there's nothing stopping you from adding on a drop-down screen and a projector for night-time viewing (assuming finances cooperate, of course).