Inkscape - transparent fill, but still occlude paths below it

I'm doing line art in Inkscape. For the sake of saneness, I need the fill of a path to occlude paths beneath it, so I just fill everything with white.

However, while exporting to a bitmap, I don't want the fills to actually be white - I just want to export the lines, with no fills.

---

Here's a example:

With fills:

no fills:

what I want:

---

I know I can replace the white with transparent in other software, but often the results are not perfect (antialiasing results in white fringing).

I also know this can be done with boolean operations on the paths, but it's not really feasible for complex work, and destroys information in the process.