I don't think you'll have trouble finding very bright projectors.
I recently bought one of these, 1300 lumens:
https://www.amazon.ca/Optoma-HD80-1080p-Theater-Projector/dp/B000R4J69K
... definitely bright enough for our needs although we use it only at
night anyway.
This NEC model is almost 3 x brighter (!) with a conventional lamp,
http://www.necdisplay.com/p/multimedia-projectors/np-m363w (3600 lumen)
Their range goes much brighter than that, though, including
laser/phosphor models:
http://www.necdisplay.com/p/multimedia-projectors/np-p502hl (5000 lumen)
--Toby