Nothing beats the sun...
That being said, the higher the lumen output, you more of a difference there will be between ambient light and projected light.
Controlling ambient light is the key to projection. Maybe invest in custom automatic blackout shades with quiet motors that can be raised and lowered as needed. Then, you just need to worry about the house and/or stage lighting.
There's also the option of LED walls, which may not be as cost-prohibitive as you think if a small pixel pitch is able to be compensated for due to viewing distance. A lower nit value can mean lower price, too.
- Jordan Wolf