Won't all greiges have warm/beige undertones? If you don't want that, maybe you could try a true gray, like BM gray owl.

Classic Gray 1548 from BenM.

Have you tried SW Agreeable Gray 7029. It is light but still has the tones you are looking for. It goes more gray than beige usually. It might be that there are flooring colors that are affecting your tones, or lighting? Have you tried changing light bulbs? LEDs are more bright white and tend to preserve true colors a bit better than incandescent? Might be a couple of things to consider.

I have Sterling from Behr that is a light gray and definitely does not look beige (I hate beige too). I plan on using it for my upstairs hallway and bathroom. I have samples painted in both places and I love how it looks in all light.

Try Benjamin Moore "Wish" AF 680, at ONE HALF FORMULA STRENGTH. They can do that at the paint store for you. Pretty darn foolproof.

I will add, that since we do not know your carpet in the bonus space, and I am fairly certain it probably is carpet?? this could be the problem you are having.............your eye sees color by comparison. A very cool beige carpet tone on your floor, can make paint appear warm by COMPARISON.

We have Benjamin Moore Abalone (2108-60) in our family room - it's not 'almost white' but it's a really gorgeous color - it fades into the background in the room but feels very comfortable and homey.