My thoughts are that your antenna theory has holes in it.
If you want it to tune on 15 meters, it needs to be a fraction of a wavelength long on 15 meters - which a antenna tuned for 10 meters has a real hard time sometimes tuning up on 12 or 15 meters.

Instead of trying to redesign the mouse trap, why not invest your money towards a used 11 meters chicken band antenna.
A antenna - regardless of who makes it or what they market it for - is a antenna.

By shortening a 11 meter antenna a couple of inches, it will work wonderfully for 10 meters.

The suggested length of coax with solid dielectric .66 velocity factor usually comes out somewhere around 18 feet long for a mobile, so working with the Smith Chart will get your SWR - line length to a better SWR, but to make it resonant - it needs to be a certain size.

You would be better off designing the antenna for 15 meters and then using the transmatch to tune it down to 10 meters then to try to use a 10 meters antenna and try to tune it up for 15. Then again, your antenna really isn't any good for either band in my opinion, due to the fact that a 5/8ths wave antenna probably works best in most situations for a base station radio for 10 meters and your antenna is way too short.

Even a 1/4 wave antenna at 15 meters would probably need to be at least 11.5 feet long, with a nominal length of about 31 feet using a vertical antenna.