Just to clarify...does the door not latch? And the gap is only on the bottom, correct?
If it's only gapping at the bottom, then I don't know why you couldn't just build a new, raised threshold. Or if that would cause issues with the floor, then could you lower the door and add extra to the top jamb? Either way, if you're not interested in removing trim, it's not going to look perfect. The best option would be to remove the door, order a regular pre-hung door, re-size the rough opening, and then install.