The Test Championship should be a true Championship, ie a Test League, based on home and away matches between every Test country in a four year period 'season'.
To make it fair, make every series a minimum of 3 Tests, and for those series, such as the Ashes, where it's more than 3 Tests, only the first three Tests count towards the Test League.
The winner will be based on a one-off Test Match final between the top two in the League at the end of the four year 'season'.