Rim Joist Insulation without a Sill Plate
That's fine. The thing you want to make sure is that you seal off any air paths from the inside of the basement to the rim joist, the reason being that if you insulate it but let humid air hit it, that air can condense on it and cause rot. That's why they tell you to use rigid foam--which is an air barrier--instead of something like air-permeable fiberglass or mineral wool batts. So as long as you make the rim joists airtight, you're fine. At that point, it's just an issue of aesthetics. Your idea of using a 1x3 there will work fine if, again, you make absolutely sure that interior air can't touch the rim joists, using caulk or expanding spray foam.