I've looked into the Solar Systems mod and it appears to be rather tightly integrated into the Any Planet Start mod, even specifying it directly as a prerequisite.
Planet Picker and Any Planet Start are not technically incompatible, but they really aren't meant to be used together. APS sets the starter planet for all players and changes a lot of things in the data phase, whereas Planet Picker performs most of its work at the runtime phase allowing each player to pick different planets rather than starting together. Because of this, the APS integration can do things like alter the prototypes of the planet you start on, which I believe is the problem here.
It looks like the surface settings for the planets from the Solar Systems mods are misconfigured until the mod itself spawns the surface, and since Planet Picker needs to ensure the surface you're starting on is generated before previewing and spawning, the invalid surface configs cause the crash.
I tried disabling the preview stuff temporarily to see if you could still spawn on the planet, thinking it might be possible to avoid this problem by gating the preview feature behind a setting, but it still wasn't happy. I'm not entirely sure what they're doing with their surface configs or why it ever has to be invalid, but it seems to be a fundamental incompatibility between how they're setting up their planets and how Planet Picker necessarily has to interact with them.
Ideally the planets' surface configurations would always be valid, and APS would not be a mandatory dependency. Not much I can do from my mod without writing a lot of compatibility code specifically for these mods that will have a performance impact, might cause issues with other mods, and will likely break the moment they make any changes to the surface configs.