OK, it took me a while (thinking slowly this morning), but I understand now what is going on. That error is misleading. It is not that there is no intersection between the body and the splitting tool, but that the splitting tool does not cut the body into two pieces. This is a limitation/the nature of the beast for Split Body. In order for Split Body to succeed, each Split has to divide the entire body. In your model, if you try to split the body using this splitting tool:

The operation will fail, because there will not be two bodies that result. Ideally, Split Body should allow the selection of multiple tools, so that you could split this thing all in one go. You could do stuff like, define sketch regions that you want to cut that cut the body all in one go.

then extrude a surface:

and use that as the splitting tool:

which will trim out one section:

You can also, I think, use Boundary Fill to do this. I need to think about that some more, and I'll try to create a video later today to explain that approach
Jeff Strater
Engineering Director