With the existing framework of &[option]=[value] for the url, I could imagine the following parameters (basically imitating the options from the link above):

and so on.

I think the default should be the small/medium badge instead of the donut (maybe a bit too big), but that for each paper.
Maybe the main style elements could be defined as option in the url (like you already use now for other aspects).
I think Altimetric might be popular enough by now (many journals have in integrated on the article page) that it find more use than the download counter.

Thanks! I just pushed the modified CSS, and quickly renaming the culprit worked!.

P.S.: Not sure if you get pinged, but I added a new comment to a closed suggestion -> https://bibbase.userecho.com/communities/1/topics/294-altmetric-integration#comment-1064

How about auto-generating a small badge from the DOI in the bibtex? Similar to what they offer for free here: https://api.altmetric.com/embeds.html