Any generic handle should work on that.
Why did they even bother putting such a complicated handle in the box.
Looking at the small white tube from the fill valve, that should be pointed into the overflow tube to refill the bowl after a flush.
The parts for the St Thomas tank come in a bag, so normally the overflow tube would be farther away from the fill valve. If you were to remove the flush valve, I would put the overflow tube on the far side, making sure that it doesn't block the bolts.
Most toilets come with pre-assembled tanks. I guess St Thomas can't pay people to do that.
The water level in the tank should be within 1/2" of the top of the overflow.
How has the flush been? Or maybe I should ask after the white tube has been positioned into the overflow tube.