Does your existing thermostat have a "fan only" position (i.e., system "off" but fan "on")? If so, I'd just use that. If it doesn't have that option, I'd pull the thermostat from the wall and see which wires it has hooked to it -- my thinking is that if your furnace will run in fan only position but your thermostat can't, you may be able just to change the thermostat and achieve what you're trying to achieve. If all that fails, you can probably wire in a switch to apply 24 volts AC (which is normally operating circuit control voltage) to make your furnace fan run on its own. And if THAT won't work, you can always flip off the gas pilot light on your furnace during the warmer months and then do whatever's needed to make your furnace fan cut on.
Basically what you're doing is a form of geothermal cooling -- the basement concrete slab is always going to be cold, and moving the warmer air around the house and getting it closer to the slab so the slab can absorb some of the heat (or, put differently, moving the air the slab has already cooled to the warmer parts of the house) is going to make things feel more comfortable.