I'm new to this so please let me know if my logic of comparing `cosine similarity` and `k-means` is incorrect

I got a set of `4 clusters` from `k-means` and now I'm interested in the `Cluster No. 1`. For this cluster, I take the `average of all values for each column` and keep it aside.

Now, I have a test sample, for which I run `k-means prediction` and I get output as `1`, meaning it belongs to `Cluster No. 1` which is good for me but my use-case here was to calculate that even if that sample didn't belong to `Cluster 1`, how close was it to falling in that `Cluster No. 1`

Hence, to resolve this I thought of doing a `cosine similarity` between my test sample and the one where I take `average of all values for each column`. Now, in this case, I get a similarity of just `5%`

I'm not sure, for my use-case `i.e. (Getting the probability/closeness of a sample belonging to a specific cluster)` which is a better interpretation for me?

I know I can use the cluster labels as `y` variables and make `multi-class classification model` but I want to keep it as `un-supervised` as possible. Please guide