Python K-Means clustering and maximum distance
Use hierarchical clustering.
With complete linkage.
Finding the true minimum cover is NP hard. So you don't want to do this. But this should produce a fairly good approximation in "just" O(n³).
This is basic knowledge. When looking for a clustering algorithm, at least read the Wikipedia article. Better even some book, to get an overview. There is not just k-means...
I would like to start by saying that my knowledge of clustering techniques is extremely limited, please don’t shoot me down too harshly.
I have a sizable set of 3D points (around 8,000) - think of a X, Y, Z triplets, for which the Z coordinate represents a point in the earth underground (negative). I would like to cluster these points using the absolute minimum number of clusters, with the following constraints:

Use the least number of clusters
All points should be included in the clustering, which means that any point should at least belong to one cluster
The maximum distance between any point and the cluster centroid (shifted at Z=0, on the earth surface) should not exceed a certain fixed distance d.

I was thinking to use scikit-learn k-means approach, by iteratively incrementing the number of clusters and then, for all points in the dataset calculate if the distance between the point and the cluster centroid (at Z=0) is less than the specific distance provided.
Of course, I am open to better/more efficient suggestions - the clusters, for example, do not need to be circular as the ones returned by k-means. They can be ellipses or anything else, as long as the constraints above are satisfied.
I welcome any suggestion, thank you for your insights.