Its not the carpet, Its the DUST!!! I would get a good electronic air filturation unit to REVOVE the dust. You can install one to a forced air HVAC system. In you have baseboard heating and no air ducts you can install room air cleaners. In addition I would get room size air purification devices in the problem areas. Then, I would put new carpet remmnents back down. YES, new carpet! You can get a 12 X 12 remmnant for less than $100. After all, who wants to walk around in their bedroom on bare concrete. I sure don't. I guarantee you would hate it! Then you might have to clean more often than once a year (not meant to be an insult). Shampoo carpets at least once a year. You may want to have your heating ducts cleaned if you have forced air. See if they make any special filters for the vaccum that trap more dust.
Or do you really want a cow barn rubber mat in your bedroom. Not much class.