Comments on your plan, and my plan
If I understand your idea, I think it would require that every HR candidate wage a national campaign? And every voter would have to choose from hundreds of candidates? Way too much like the CA recall to suit me.
My idea: Leave the HR as it exists today. Make an apportionment robot that is free of all bias (and doesn't know where the partisan boundaries exist). Whenever a census is completed, run the program and do an automated apportionment for the entire nation. No muss, no fuss.