I suspect that the problem has to do with the MIDI channel, because the software is probably displaying all of the notes from a given MIDI channel on a single staff.
You might be able to adjust the score in the software, such as by adding a bass clef staff, copying the notes from the treble cleff staff and pasting them in the bass clef staff, and then deleting any notes from each staff that you don't want, such that the notes are split between the two staffs the way you want them.
But if you want to split the notes up automatically, you might be able to do that by turning on your keyboard's Split function so the right-hand keys will be transmitted on one channel (probably channel 1) but the left-hand keys will be transmitted on another channel (probably channel 3). You'll probably want to set the left-hand voice to be the same as the right-hand voice, so it sounds like you're using one voice across the entire width of the keyboard. You'll probably also want to move the Split Point to somewhere near Middle C. Another option might be to put the Split Point around F# below Middle C and raise the Octave setting on the left-hand voice to +1, so that you can play a few keys below Middle C with your right hand yet have them go to the treble clef staff, and play a few notes above Middle C with your left hand yet have them go to the bass clef staff, although that would mean having to play the keys an octave lower than usual with your left hand to compensate for the raised octave.
Of course, you would also need to set up the software so that the right-hand notes on channel 1 or whatever are displayed on the treble clef staff, while the left-hand notes on channel 3 or whatever are displayed on the bass clef staff. I have the free version of Finale NotePad, so I'll look at it later to see if that's doable and what the steps are.
