Honestly my last TV purchase was a 65in Hisense and was under $1000 after taxes while on sale, with quantum dot and a few other things so I kind feel like my opinion on TVs is not up to snuff for what you might be expecting however a couple points for thought.
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65in OLED tvs within your budget and money to spare I wonder if the image quality of the OLED might not trump the 10 inches depending on room/useage?
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Also depending on room/usage I suggest seeing if you can't find a "ultra short throw projector" that is on demo at a place like visions.
These newer style of projectors work in a lit room. This linked one is a 100in laser projector with screen, the laser bumps the price but LED ones exist too at cheaper price points but this is a complete package with the special screen included. Normally blows past your budget but sale price right now brings it close. I feel like they are atleast worth looking at if you can if size is really what your after.
www.visions.ca
Not that I care for this video in general he explains how these are working in a lit room. I feel like they at least worth seeing in person if you can.