Geographic transformations are designed to convert between two geographic coordinate systems only, not between two projected coordinate systems. The X/Y/Z axis translations are offsets in the 3D geocentric frame (X and Y are in the equator's plane, Z is polar axis), which is why the results aren't correct.
Instead, I think you should try to make a custom projected coordinate system for your local grid data. Based on what you've posted, it looks very possible. You can't use the standard UTM definition with it, because the local grid is definitely not in UTM.
The easiest way to do it is to remove any existing coordinate system from the local grid data--"clear" it using the data's property page in ArcCatalog.
Add it and the reference data to ArcMap.
Set the data frame's coordinate system to your calculated parameters. You may need to adjust the parameter values, but ArcMap will convert the reference data to the new system automatically and you'll be able to immediately check the fit. Use the UTM zone definition as the basis, but change these parameters:
false easting: 4254.108m
false northing: -5717240.125m
Scale factor: 1.0000768092 (0.9996 * 1.000477)
other parameters stay the same
You probably also need to multiply the false easting/northing values by 1.000477, but try this first and see what it looks like.
Hope this helps,