Inkscape's pdf generation is usually very good so I wouldn't go complicating matters with additional software/tools.
Sometimes a file benefits from the objects being ungrouped. Sometimes you'll find that a slightly open object won't export happily and sometimes flipping the direction of a path helps.
I've had a quick play with your file and I'm stumped as to why it won't export the simple ovals. If I were you I'd quickly re-add these elements as a test oval I added exported fine.
I notice that you're using filter effects - bear in mind that these result in non vector elements which may cause you trouble. They get rasterised at a resolution that you set on pdf export.