A measurement would have to be taken to see if then air purifier is generating any static charge. I would doubt there is any charge generated by purifier if it is a UV light type purifier, and if any it would be very low charge. Some air purifiers use electrostatics to charge the air to aid in the removal of impurities air these types of purifiers may cause the air to be more charged. Air purifier are used to CLEAN air, although there will be some filtering properties to these types of units.
You mentioned that you have a desktop ionizing fan at your workbench. This device at the work site will control most static charge within the working volume of the ionizer, as long as the ionizer is operating properly with a properly balanced output, which can be verified using measurments. I would recommend using passive ESD control measures as well as the ionizer (wrist straps, properly grounded work surface, removal of all insulative materials from workspace, ect.). Taking the time to ensure the work site is as free from charge as possible will prevent damage to the computers you are building.
As far as the dust; control of the static charge will help in controlling some of the electrostatic attraction of the dust from the room, but a better route may be that you establish a temporary “clean room” setup to keep as little dust in the working area as possible. I’m not suggesting spending large amounts of money to build a certified ISO class clean room, but an area that can be isolated from the rest of the room with low charging (ESD safe/low charging) plastic walls, and a fan/filter unit blowing air from the top of the structure down over the work surface may be an effective way to keep the work area clean. There are several companies that sell these types of setups, or you could build your own for the possibly the same cost of the air purifier (Internet search would give you ideas). Just a suggestion to get you through this crazy time we are all going through.