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It's not possible to really answer these types of questions accurately without seeing the room in person, that said you could just buy a really bright projector somewhere that has a return policy to try it out.
As far as the screen material, some type of retro-reflective screen or gray screen might help for ambient light, but it depends on the angles that the viewers will be seeing the screen from (if at too much of an angle it could hurt more than it helps). If they are viewing at large angles, I suppose a regular uniform gain screen at around 1.4 gain could help by increasing brightness. You can also just paint a special type of screen paint onto any surface and use that, you will find more info for that in the DIY screen forums.
How big does the screen really need to be?
You could just wall mount an 80" Sharp LED, they cost about $3000 online. These handle ambient light much better than a projector would. Though the Sharp may have viewing angle issues of its own, but I don't know.
The reasons it's not possible to give an accurate answer online:
Depends what you mean by well lit and depends how big of a screen you want, as well as how much you are willing to spend.
Also depends on what angle people are viewing the screen from in relation to the angle of the lighting in the room.
Like if the room lights are between where the viewers and the screen will be, or if the lights are hitting the screen directly,
Well lit doesn't mean much to me, but I'll assume well lit in this case means about as bright as a department store.