Project UTM WGS - Lambert Conformal Conic

Hello dear members in this forum!

I'm writing my Bachelor's thesis and I have a severe problem with two shapefiles using different coordinate systems and really hope you might help me!!

The first shapefile is in UTM WGS84 Zone 32N and the other one in a custom Lambert Conformal Conic with European 1950 Datum.
I want to project the UTM-shape into Lambert, but I find no way at all in which the two shapefiles go congruent.

I'll give you the parameters for the Lambert coordinate system for better understanding. The UTM-System is common (WKID: 32632).


ED_1950_Lambert_Conformal_Conic
Authority: Custom

Projection: Lambert_Conformal_Conic
False_Easting: 0.0
False_Northing: 0.0
Central_Meridian: 10.5
Standard_Parallel_1: 48.66666666666666
Standard_Parallel_2: 53.66666666666666
Latitude_Of_Origin: 51.0
Linear Unit: Meter (1.0)

Geographic Coordinate System: GCS_European_1950
Angular Unit: Degree (0.0174532925199433)
Prime Meridian: Greenwich (0.0)
Datum: D_European_1950
Spheroid: International_1924
Semimajor Axis: 6378388.0
Semiminor Axis: 6356911.946127947
Inverse Flattening: 297.0


I tried all of the geographic transformations which were suggested in the "Project"-tool, but every time the two shapefiles show an offset to each other (about 300m). I also tried to research transformation parameters for my region to define a custom geographic transformation, but that didn't help as well. The two shapefiles are located in the Austrian alps.

So I would be so happy if anyone of you had an advice in this issue. Maybe I'm overlooking something here.
Thank you very much in advance!

Greetings,
Caro