European soil texture dataset according to USDA classification
After a lenghty unsuccessful search, I found an option to do my own mapping. There are global datasets with the percentages of clay, silt and loam available from the Land-Atmosphere Interaction Research Group at Sun Yat-sen University and I used the R soiltexture package for the classification. I ended up making a dataset covering my domain of interest in 2 km resolution. I attached my map and the original map mentioned in the question for comparison (some of the colours are a bit off, but they should more or less correspond to the same category in both images). If anyone is interested in this data, feel free to let me know. My custom mapping.

Original dataset from the European Soil Data Centre (ESDAC)