Sorry I am short of time, but hopefully I can help:

Instead of using negative modifiers, just load the full part as it is. Then rotate it (Usually ±90 degrees) around the x or y axis and use the z cut tool to cut the part in half. Only keep the part you want. Reverse the initial rotation and the object is now cut on one side.

Redo the steps for all 4 sides.

Edit: also this might help (dont know, if it's an beta option): https://help.prusa3d.com/article/cut-tool_1779

Could it be that you have a brim or skirt set for around the hole part that steps out of the build plate?

OpenSCAD is particularly good for jobs like this. You can import an stl and do an intersection with a "cube" to get the part you want.

OpenSCAD is script based, so does have a little bit of a learning curve, but it's an excellent tool to have in your toolbox.

Disable Auto center then drag it around.

Edit: my reading comprehension kinda sucks sometimes. The original comment shall remain as a monument to my eagerness to help, but unwillingness to fully understand the problem first. :/

Maybe try pulling it through the build plate? I've done that but I'm not sure if it complained about the non printing part. Maybe also try cutting the model with the slicer?

Stop using the slicer for part modification would be my advice. Sure, it CAN be done - but you're running into the reason most people don't.

Most slicers will allow you to drop the part below the build plate and cut that area off though. That's about the extent that I would trust the slicer to modify it.

Cut it up before importing in the slicer?

I second the other comment it’s best not to handle it in slicers.

Rather, I’d suck it into your favorite modeling software, then create a sketch with a cube that fills the space you want to cut (so one face is the cut-plane and the rest of the part is inside the cube.)

From there, duplicate the part you’re trying to print and use the cube to “intersect” the part- which makes it so that copy, only the stuff inside the cube exists. On the duplicate, you can then use “cut” to remove the bits you just created.

In Fusion 360, the function your looking for is “combine”, you don’t need to duplicate the mesh part just toggle “keep tools” (I’d also suggest “new component” as well.)

While you’re in there… I would create a sketch off the net cut plane (either side,) and then create registration posts. Shallow is just fine, but you want a something tapering to a solid point so that as you’re gluing up, it aligns itself. (A shallow pyramid also works, whatever angle youre printer can handle overhangs is fine.)

Inside, you’re joining the extrusion the other you’re cutting (and creating a void.)