A 3,000 lumen projector is good enough for a normal room with light and higher for brighter areas. You can go from there to see what is available in your area at what price but your budget is probably not high enough to get a projector that is good at that brightness. They tend to be near the 1,000 mark to start with and go up for those with good resolution.
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.