Large TV for Well-Lit Room
Yours is a tough situation for any video display! My first suggestion is to get motorized blackout shades for the window behind the TV. That much light coming from the same area as the TV will seriously bias your vision, causing you to perceive much less low-level detail and contrast in the video image.
These days, most LCD TVsincluding those illuminated with LEDshave shiny screens, just like virtually all plasmas. This helps create deeper blacks, but it also reflects light from objects in a well-lit room, which is especially evident in dark scenes. The only type of display with a matte, non-reflective screen that produces enough light to stand up to strong ambient light is a rear-projection TV, and the only manufacturer making such sets today is Mitsubishi, which offers RPTVs with screens measuring 73, 82, and 92 inches diagonally. This is what I recommend for you.
It sounds like you probably have a big enough room to accommodate a large RPTV, even though it's quite a bit bulkier than any flat panel. And RPTVs are relatively inexpensive for large screen sizesfor example, the top-of-the-line, 92-inch WD-92840 (seen in the photo above) lists for $7000, but I've seen it online for $3000 to $4000, which is right in your budget range. The step-down WD-92742 lists for $4500 with online prices in the $3000 range. If you don't want one that big, the 82- and 73-inch models are much less expensive.
You might also consider the 75-inch L75-A94 and L75-A91 LaserVue models, which use lasers instead of conventional lamps, so you won't need to replace the expensive lamp after a few thousand hours of use. Both models list for $6000, but I've seen them online for $2300 to $4500.
I can't say how well any of these models perform, since we haven't reviewed any of them. The last RPTV we reviewed was the 73-inch WD-73837 back in 2009 (see review here), and it received very high marks, so I would expect the current modelsat least those with a conventional lampto perform very well. I don't know about the LaserVue sets, since we have never reviewed one.
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