You can't test your swatches on that current yellow wall color! Your perception of the new colors are influenced by that background color.

Prime the walls white and test again.

We have ballet white in our game room and I don’t think it reads ‘too’ white, more warm neutral like you were looking for. It really looks a little beige against our white trim. Now, you and I might have very different definitions of ‘warm neutral’ of course so keep on sampling until you find what you like.

Ps. I agree with above, your current wall color will skew your perception of the sampled colors.

I'd probably pick an actual color for the walls - blue, green, yellow - rather than gray. If you do go with grey or white throw in plenty of color with accents and murals.

Any color you like - there is nothing inherently boy or girl about any color unless you make it so. A pale warm blue-green (tending toward turquoise) can work as a background color for lots of other colors.

Look at Navajo White; it is different across manufacturers (some more yellow undertone, some more green and so forth), but tends to be a complex warm off white that works with many other colors. And remember that if you find a color you love from another manufacturer, BM can probably make a close match.