Good evening MM.
As someone who has carried a video projector from Stavanger to Mossel Bay and Bristol to Ulsan, and many places in between as part of my former day-job, a darker room was often better for displaying pictures, ie photographs of some odd items of equipment.
It was not necessary to have the room dark like a cinema, a good compromise was to have no lights shining directly on the wall used as the projection screen.
Another thing to consider is the size of the projected image, the bigger (wider and taller) the image, the lower the brightness for the same lamp power.
In the case of Mrs MM and her fellow inmates, I guess that your lyrics can be simply black text on a white background, or white text on a black background, for maximum contrast and best readability.
One thing which occurs to me, and I don't know if this exists as a simple piece of software and readily available...
Something to display just one or two lines of text at a time with a jolly bouncy pointer highlighting where you are in the song.
This may be difficult to sync to your playing though.
As for the video projector itself, when I bought mine about 20 years ago, there were cheap ones and there were good ones, mine cost me just short of £1k, cheaper ones were available for £200 or £300.
I never regretted the choice that I made.
My projector has an incandescent lamp, but the newer projectors have LEDs of course.
The only LED projector that I have seen close up is nice and bright, but I don't know how much it cost.
My experience for what it is worth.
JimB