For the task, it is of course important that you have a sampling of profiles that is dense enough to ensure that the result becomes precise enough. If you have a 1,6m DEM and profiles pr. 100 m. then your profiles may be more precise, but your overall uncertainty will be huge.
If you have sufficient data in profiles, then you create points along your profiles with the z-value and interpolate a raster DEM based on the data. Make sure to use the same reference grid as your existing DEM. Then you do a simple condition statement afterwards, where you patch the two together.
The choice of interpolation method is optional and should be based on analysis of data, but generally i would go for Krieging.