I think there may be a basic misunderstanding here. Ignoring details such as losses through the actual optics, the image brightness is a function of the image size, not the throw distance.
If you have a fixed lens the projected image brightness does indeed drop as the throw distance increases, however that is because the image area over which the projector output is spread is also increasing. In comparison, with zoom lenses there is no longer a direct relationship of image size to throw distance, and thus of image brightness to throw distance. If you have X lumens over a A' by B' image then it does not really matter if the projector is 10' away or 100' away, either situation is the same projector output over the same image area and thus the same image brightness.
You can also look at this the other way. Pick a throw distance, then adjust the projector zoom and as a result the image size changes. Same projector and thus the same light output, but over a varying image area and thus a varying image brightness. It is the area over which the light from the projector is spread and not the distance from projector to screen that matters.
On a lesser note, be wary of assumptions regarding where a projector can or should be located, different projectors and even different lenses on a projector may have different requirements in terms of where the projector should be located relative to the projected image and also in how much compensation the projector and lens can accommodate if not in the ideal location.
Going back to the OP, consider this. One factor for the projected image is the image brightness or how bright white can be. However another factor is how dark black is in the image. The difference between those is the image contrast and insufficient contrast is what relates to an image being 'washed out'. You can try to improve the projected image contrast by increasing the white level by using a brighter projector but you can only go so far before that is impractical or plain difficult to view. In most cases the better approach is to try to reduce the black level, which is generally controlled by the ambient lighting on the projection surface.
Rear projection if often a good alternative, however it does require a proper screen surface and sufficient space behind the screen. You can shorten that distance with very short throw lenses, however such lenses often have tradeoffs in terms of hotspotting and in where the projector must be located and how much that can vary. Mirror systems can also be used to 'bend' the light path reduce the physical depth required but add cost and require that nothing else be in the 'folded' light path.
For front projection do not worry too much about the distance to the projector or projecting through other lights in terms of the image brightness. To be technically accurate, long throw lenses due typically result in a bit less output from the projector but that is due primarily to losses within the additional elements in the lens. But, if you have a projector that can be used with different lenses or for which an extender is available then you may be able to get appropriate optics to allow the projector to be further away such as in the Tech Booth. Of course that may create other issues regarding noise, heat and power in the tech booth, but it may at least allow that option.
------------- Brad W.
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