Simply having a bright projector will not make it work well in a bright room. Projectors require a dark room because they can't projects blacks (there's no such thing as black light).
On a white screen with the lights on, all the dark detail will be lost and the image will look washed out regardless of how many lumens it puts out.
To use a projector with the lights on you want a dark colored screen material - sometimes referred to as ambient light rejecting screen or just "alr screen".
You can see my old black screen in this video:
Even with an ALR screen you won't get great performance in a bright room. You can achieve OK performance in a dimly lit room.
On your budget you should forget the led thing. Your options are limited.
You can try and find an old Sony Dynaclear screen on ebay. These were an 80" portable ALR screen and usually sell for around $200. Then you can pair it with an old used ProjectionDesign F30 series (also off ebay).
The ProjectionDesign F30 series are dual lamp projectors ranging from 3000-7000 lumens. If you're lucky, you might be able to grab a working one for less than $500 with a lens.
Your other option would be to grab a cheap ultra short throw projector off ebay. These sell for as little as $150. They usually aren't great (in terms of quality) but, if used with a gray rear projection screen, they'll work OK with the lights on.