The problem was always that I didn't know where to get a method that can handle all data types. You may have numerical variables such as leaf length, where it appears obvious that 5 cm is twice as far away from
The R package homals can do a homogeneity analysis, a generalised form of PCA-type ordination analysis, with such mixed datasets. But that is it, it will give you a 2d or 3d plot. It will not do clustering.
This week some concerted googling finally produced the solution: The R package cluster (unsurprising, I know) and its functions daisy and pam (of which I had not heard before). The name doesn't have anything to do with the Asteraceae family, the distance functions in cluster rather appear to be named after female names. When presented with mixed data, daisy defaults to the Bower metric as a distance measure, which is exactly what I wanted.
Because others might be interested in doing non-hierarchical clustering with morphological data, and because getting this right was quite a struggle (as so often with R), I thought I would post a how-to.