Improve DTM under forested areas without smoothing?

It is not possible to directly improve the DTM without carrying out some sort of smoothing or interpolation. Auxiliary data is needed or the DTM needs to be rebuilt.

To improve the assigned values of elevation in the part covered by forest, return to the raw LiDAR data and test alternative ways to classify ground points and generate a new DTM. For example:

Determining bare earth DEM from unclassified LAS file?

If there is not a satisfactory method to improve the results, you will probably need data with higher resolution in such parts of the terrain. It is like you said:

I get similar results when working with Slovenia LiDAR las data in areas where they scanned 2 points per square meter. In other areas where it is 5 points per square meter DTM is good even in heavily forested areas.