... my first choice would be a 3-4 day trip and if that includes getting to the top of something great but if it is just spending a few days in pretty country and exploring around that's perfectly fine too.
You can ask if there are any extra overnight permits available for any of the area trailheads (including the main trail) when you arrive at the Visitors Center in Lone Pine. It costs you nothing to ask, and you may be happier as a result.
Edit - we're trained, but I guess won't really know until we try just how well is well, right?
Right. Any number of things can happen, but your odds appear to be pretty good.