Although no final route has been selected, we can make some preliminary estimates of cost based on a detailed study by the Washington Department of Transportation.
WDOT studied in detail 36 projects, of which 13 cost in excess of $10,000,000 per lane mile. The ones costing less than this were typically simple projects with few interchanges and/or bridges. They ranged from $1M per lane mile for simple widening (adding a lane) of an existing expressway, to a staggering $188M per lane mile for the Boston Big Dig.
An inspection of the projects they studied indicates that $10M per lane mile is a reasonable estimator for the I10 Bypass project.
On this basis, the approximately 210 miles of 4 lane highway (840 lane miles) would cost $8.4B dollars, and the costs of highway construction have been increasing faster than the general rate of inflation (the WDOT study was 2004).