This should work
There doesn't seem to be a requirement that the invented weapon remains a valid choice for creation. You are correct that, using a Shifting rune, you can reshape a Simple or Martial weapon to be an Advanced one (assuming it meets the Handed-ness requirement).
Unconventional Weaponry will allow you to treat an Advanced weapon that meets the Feats restrictions, for Proficiency, as a Martial (not Simple, but that's fine for you) weapon. So now you have your inventor-modified weapon in the shape of an Advanced weapon for which you have your current weapon Proficiency for.
It could be argued, from a (probably excessively) legalistic point of view, that altering the form of the Innovation to be an invalid weapon prevents you from accessing the Innovation abilities for it because it's 'no longer a valid target'. For this reason, I'd recommend communicating your intent to a GM where you want to use this build; I wouldn't expect a ton of push back (especially because Advanced weapons don't tend to be significantly more powerful than Martial options) but it's always best to be on the same page.