No way of creating a window which does not accept mouse inputs/events (click-through) but isn't totally transparent/invisible(?)

[MrMeugh]

I'm trying to make a "desktop pet" style program in which a graphical element is shown on screen, always on top and not (visually) in a window. Someone using this program would probably continue doing what they usually do on their computer, but with this added thing on top. Ideally (really, necessarily), this graphical element would not prevent the user from using programs underneath it. As far as I've seen, the only function that allows something like this (mouse pass-through/click-through) in SDL3 is SDL_SetWindowShape, but I can't use this for my program as transparent areas will not show anything rendered to them. SDL_SetWindowShape with an empty image (apart from 1 pixel in the corner) gets me very close, in terms of a full screen, transparent, borderless window with click-through; except there's nothing on it. It seems like what I need is a function like SDL_SetWindowShape via which mouse-transparent areas could be defined, but which doesn't affect rendering.

Is there a way to do such a thing in SDL3? If not, would it be possible to modify an existing function to do this? I'd like to avoid using other libraries and the windows api, if possible, although I am willing to modify SDL3's source code for my project.