How can I find which client(s) is causing my distribution update to fail?

I doubt you'll be able to resolve this if this is true "I don't have backend server access to the management points, or the DPs".

Can you access the distmgr.log on the site server? If not, then you will need to escalate the issue to someone who can.

This issue is nothing to do with the client so I would ignore other answers advising to look at the clients. This issue is caused because the Site Server cannot copy the files from your source folder to the distribution point.

If you can't access the Site Server logs, one thing you could try to eliminate it being down to your folder structure being too long is to zip up your package, deploy it, and unzip before installing at the client end.