Why not drop the road bed a foot?
^ That spot's already a natural depression, so it would impact drainage. Would also impact the railroad bridge piers, the adjacent intersection, and potentially the historic brick building in the southeast corner (where the bridge camera is located).
Most of these city underpasses in North Carolina (like this one at Gregson Street) have utilities in the street that would need to be lowered before the road can be lowered. In the case of sanitary sewers, they may also need additional pumping capabilities. Worse, the drain sewers may not be able to be lowered without lowering everything downstream.
During the ill-fated (then) TTA Regional Rail Program (almost 20 years ago), we got to work closely with the NCDOT Rail Division specialist on grade separations. He gave a great history of the problems with the existing railroad underpasses. In a nutshell, these old underpasses were placed on "main roads" that almost immediately became obsolete because of the advent of larger/taller trucks. Only a few were large enough to continue to be utilized, and the rest of these were bypassed. So you've got a situation where these little underpasses are on (now) minor streets that are not deserving of the necessary investment. Because they are mostly used by cars, it is hard to justify closing them altogether (since it is often safer than routing the traffic over an adjacent grade crossing).
By the way, that fellow I mentioned is now the Director of the entire NCDOT Rail Division. Looks like he is doing well.