Forgive me, but "do this in an RGB document" is not a workflow. No need to comment again. I'll keep working on it myself. And if I figure it out I'll post my solution.
By @johnm27994604
It won't work in CMYK. Period.
When defining your colors in HSB you are using a very large color space. In an RGB ocument those colors are preserved. In a CMYK document they get converted to CMYK the moment you lift your finger from the mouse button. This is a CMYK color space inside an RGB color space. There might be slight deifferences between this plot and the specific color spaces you are using, but basically this is it:

So you are difining something outside of the small space, it will automatically be moved to the inside and of course changed.
Next thing is that you are opening this in Photoshop (or placing, doesn't matter. I presume your Photoshop document is in CMYK as well. Trouble is that those blending modes work best in RGB. In CMYK they will work differently or not at all.
So good luck with your project. Maybe you want to read a bit about color and color management.
Works in RGB https://youtu.be/oN1yKAn9EO4
I finally got it to work, and yes, both docs, in Photoshop and Illustrator are set to RGB. You're right, when in Illustrator the Document Color Mode was set to CMYK, using the HSB sliders wouldn't stick. But I do need to learn more about color, particularly about how the warnings work (Out of Web Color Warning and Out of Gamut Warning)
The solution, which is imperfect, I think because of antialiasing, requires that the Brightness AND Saturation be equal, letting the Hue be the only variable. Which is interesting, because I thought these three would be independent variables, but Saturation and Brightness both affect to Value. Not sure why.
If you look at the gray png you can see the contours of the shapes. Which means that the antialiasing must be adding brightness or contrast, just a little. I see tutorials online for turning this off, so I'll have a look at these.
Thanks for your help!