I see that you put a +towgs84 parameter in your destination crs (WGS84), but it should be included in the source crs or in the transformation (as dx, dy, dz parameters), not the destination crs.
If I transform from:
+proj=lcc +lat_1=44.33333333333334 +lat_2=46 +lat_0=43.66666666666666 +lon_0=-120.5 +x_0=609601.2192024384 +y_0=0 +ellps=clrk66 +towgs84=-8,...
I used the proj.4 declaration provided by JohnGom here:
Proj.4/GDAL/QGIS Transformation between CRSs that are defined the same
+proj=longlat +ellps=GRS80 +towgs84=-0.06155,0.01087,0.04019,-0.0394924,-0.0327221,-0.03289790,0.009994 +no_defs
You may also need to modify other projections, depending on your needs. For example, I want to convert from GDA94 to ...
The first one "proj4js" is an adaptation from the second but not a port (same API in a different programming language)
The second PROJ (C++) is used directly or by program like PostGIS or GDAL to manage projections
Both take as an input mostly the same definitions for projection. You can confirm it with "Export" section from website epsg.io (an example ...
The main problem is in the image warping parameters, they are not giving you the right gridded image.
About the projection itself, you are using the Lambert Conical projection in the right way. You have just not information about the datum transformation.
A datum transformation is going to be necessary if we want to compare this map with one based on ...
Looking at this page which proposes this proj string for this Italy South grid:
+proj=lcc +lat_0=39.5 +lat_1=39.5 +lon_0=14 +k_0=0.99906 +x_0=700000
+y_0=600000 +ellps=bessel +units=m +no_defs
Seems to indicate you are on the right track, that may be as accurate as the projection gets?
The -ct parameter of ogr2ogr is used in the same way than all the other parameters that takes their value from a string. Just add -ct and then the string to use, enclosed between double quotes. As documented in https://gdal.org/programs/ogr2ogr.html you need GDAL version 3.0 or higher for usint this option.
I tried you pipeline for the states.shp shapefile ...
As a quick answer, most transforms are more complex than a scale/translate/unit, so you are looking for exceptions. The distortion come from the change in datum (that is the transformation, from lat/long to lat/long) and from the type of projection (from lat/long to XY).
Having the same (or rotated around the vertical) datum is a prerequisite but, if you ...