I got a working solution! I combined these versions:
BeyondHome 1.5.3 from @ballisticfox0 (thank you for your help!) https://github.com/ballisticfox/BeyondHomePlanetMod/releases/tag/1.5.3
Kopernicus v199 which just released, as of this writing https://github.com/Kopernicus/Kopernicus/releases/tag/release-199
Parallax 2.0.6 https://github.com/Gameslinx/Tessellation/releases/tag/2.0.6
I did not install the "ParallaxStockTextures" pack as mentioned by the BH readme. I did install the "ParallaxStockScatters" pack. There is a warning popup at launch about the missing terrain config, but ignoring it seems harmless.
For diagnostics: since my last post I've been varying the version of Kopernicus in combination with Parallax 2.0.6 and BH 1.5.3. Because @R-T-B just released a new version of Kopernicus, and now everything works fine, I think that was the problem causing my "infinite loading spinner" error. And because the Parallax 2.x version is a lot better than 1.x, I'm gonna abandon figuring out the black squares problem.
I hope this helps other folks looking to play Beyond Home! Thanks to everyone who makes these amazing mods. KSP 1 continues to give and give and give!