The H section provides the needed increase in height, and will be stiff for and aft, but not so rigid side to side
I would suggest welding in some struts at the ends to turn it into two back to back boxes, plus some struts at the sides (Just narrow enough o allow access to the fixings )
In this way the stiffness will increase, and any distortion from welding should be taken out when the top and bottom faces are machined.
You could even take a skim along the top of the outer edges to use as a datum face, if thought likely ever to be needed. (For Vice alignment? )
Howard