An Argentine snaffle is a Western Pelham, yes, but it is certainly not designed exclusively to be ridden with two sets of reins. In fact, I don't know a soul who uses two reins or roundings on an Argentine. You simply have the option of snaffle control, if you wish, but it's not the greatest anyway since the ring is so small.
Leave the curb on as it is. Attach to the bridle with the bit hanger in front of the curb, like you do with your Tom Thumb. And hook your reins to the bottom rings, leaving the middle ring bare.