Perhaps they should reevaluate a cut-and-cover tunnel along the proposed Connector alignment as opposed to the Thruway alignment. A cursory look at imagery and topography suggests that they should be able to build such with minimal impact on the northbound Thruway (most important side for hurricane evacuation).
Not feasible IMHO, Froggie, for the following reasons:
1) You'd still have to take out the SB Evangeline Thruway roadway, Mudd Avenue, Simcoe Street, and a lot of the Jefferson Boulevard cross section, in order to build both the trench and the cap. Way too destructive during construction, and potentially prohibitory for the neighborhood immediately west of the Thruway.
2) The plans for the existing Evangeline Thruway with the proposed Connector project is to reduce their laneage from 3 lanes to 2 lanes in each direction and incorporate "Complete Streets" design to include pedestrian and possible bicycle access. The direct interchanges with Second/Third and Johnston would support that option, since it would remove the bulk of the heaviest traffic away from the Thruway while providing improved access to these main arterials. (In addition, it is assumed that the Connector freeway would take on the bulk of any hurricane evacuation traffic. Going to a depressed or capped freeway would take away those opportunities, since the Evangeline Thruway would probably still be used for direct access to the main arterials. Also, more traffic would mean more of a noise impact, especially to Sterling Grove District.
3) The current alignment between Jefferson Street and Taft goes through property that once had the old Southern Pacific railroad yard and roundhouse, which has been known as a potential hazardous waste site. One of the main concerns of opponents of the current plan is that the pilings required for construction could penetrate into the clay sands where Lafayette's drinking water are stored (the Chicot Aquifier). It's also the reason why the current alignment between Jefferson and Johnston is now at grade on fill rather than elevated. I'm pretty sure that a depressed or capped alignment would present an even worse threat of a breach unless the site was thoroughly cleaned up.
4) There is a coulee ditch that runs across the Evangeline Thruway just north of and parallel to the L&DRR spur crossing between Willow and Mudd that essentially eliminates a depressed freeway south of Willow. If you retained the elevated highway from Willow to just south of the L&DRR, you would create an excessive gradient between there and Mudd Avenue in order to transition to a depressed or C&C section. (The same would be true as well between Johnston and Fourteenth/Taft Sts. where the freeway alignment rejoins the Thruway ROW, since an elevated freeway is required to allow the SB Thruway roadway to pass underneath, and to overcross Pinhook Road.)
5) A trench/cap would all but wipe out the Jefferson Blvd underpass of the BNSF/UP railroad, and eliminate the possibility of grade separating the Second/Third couplet and Johnston St. rail crossings.
6) If you notice, the original C&C proposal contained a partial interchange with the Thruway accessing Mudd Avenue that was proposed to be the main access from the mainlines to downtown to/from the north. Partial interchanges like that are now not considered kosher by the FHWA, who recently revised their standards for interchanges and access in urban and rural areas. Of course, you could eliminate those ramps and use Willow Street and Johnston Street (or even Willow Street and the Surrey Street/E. University Avenue interchanges) for accessing the core area by using the existing Thruway lanes as virtual C-D roads...but that would still mess with the CSS plans and "Complete Streets" proposals.
tl;dr....much more disruptive and more costly with no real benefits as compared to the current proposal.