So if I tie into the HVAC return, it will obviously draw the hot air out of the closet when the HVAC system is running (heat or AC), replacing it with fresh air from outside the closet through either a passive vent, or the air gap beneath the door.

When the HVAC system is NOT running, however, are you saying your additional fan is installed in the same air return duct, and pushes hot air out of the closet, and into the air return ducting as well? I am a bit confused on where that air will go when the HVAC fan is not circulating air through the close looped return/supply system. Will it end up being displaced back into the home through the HVAC supply ductwork? I understand the pressures will be very low compared to running the HVAC, but that is what it seems like would be happening in that scenario.

I am not sure why I "should not" vent into the attic (or to outside). My attic is vented to outside already. So I am not pressurizing the space at all. I understand that I would be dumping conditioned air into the attic/outside, possibly decreasing the overall efficiency of the HVAC system by a small margin. But this is the same thing that is happening when you run your bathroom fans, although on a temporary basis. Dryer vents are the same thing as well. Dumping hot air outside. The difference being that bathroom fans and dryer vents are not running 24/7, whereas my closet fan will likely run 24/7, at least in the summer months, even if on a thermostat set at 90-95F.

Active thermal suggests all of the above mentioned methods of exhausting air in their instructions. They mention exhausting into attic/crawlspaces, outdoors, HVAC return ducting, adjacent rooms, etc.

I could easily push the air from the closet into the laundry room. I could easily vent it to outside. I could easily vent it into the attic. I could fairly easily tie into the HVAC return. It would just require some kind of "T" splitter.

I will keep looking at options. But for this case, easy and cheap and fairly temporary is the goal....I will not be in this house for more than a couple of years anyhow. :)