Thanks for sharing this interesting problem. A few things aren't clear from the discussion that may be relevant. These may point to the dominant forces on the culvert, which in turn would suggest how it will deform, and then fail. This will then suggest what you should be measuring from which you can find the best ways to measure it.
As others have pointed out, you first need to identify the purpose of these measurements, to provide an improved understanding of the failure mechanism, or to preserve public safety. You seem to be reasonably assured of the latter.
What kind of depth of cover have you got over the roof? How high is the entire embankment?
Is this a "rigid" (cast in place) or "flexible" (precast) culvert?
How was the culvert installed, in a trench or projecting and covered with an embankment? Do you have access to records of the backfilling sequence and methods?
Is the material beneath the culvert relatively rigid or compresible, in comparison with the fill around the culvert?
Could the culvert be bending on its long axis, i.e. could the embankment be settling and pulling the culvert down with it? The description of the vertical cracks at the base of the wall suggest this, particularly if they are more predominant near the road centreline. In this case, longitudinal profiling would be suggested.
If you think this is a "wall" problem" and you've got the money, one of those Bassett convergence systems (also described at Slope Indicator)should be able to provide you with a good cross-section of the deformed culvert. If you have access to pretty reliable records of the structure, you should be able to back calculate the pressures on it.
Last of all, you have indicated that this culvert is scheduled for removal and replacement. This suggests a rare opportunity to load the culvert to failure prior to removal. In that case, you would provide a great service to the understanding of the behaviour of buried structures if you could test it to failure and publish the results.