One curve multiple objects

Maybe not the right terminology...

I have a cartoon character as a bitmap.

Run it through trace and I now have a vector version of the image.

Examining the vector several parts of the cartoon that, to a human eye, are supposed to be separate, are joined with a tiny bridge.

My normal workflow for these is to place a rectangle over the bridge, select the rectangle and the part that has the bridge, then use 'Back Minus Front' (I love that tool).

I now have separation and just need to tweak the nodes. In most cases, this is all I need, until today.

The two 'separated' items are still one curve and I need to convert these two items/objects/shapes into truly separate 'things' for cutting (they will be different materials).

Instead of the back minus front method, I've tried break apart (done twice) and then move the ends of the curves to autojoin, with the same result.

It's obviously me or my bastardised workflow, but would some please tell me the next step?