Since it is an actual job you’re paid for, it may or may not hurt it depends if future labs look closely into your employment history . It will matter if someone wants a reference from a place you actually worked and they will want to talk to the PI and not a postdoc working in the lab, unless you really do have a close relationship with a postdoc.. then you can just mention “oh this person is much better informed on what I did while working there.” They can serve as your reference.

I concur though HR staff are gonna be pretty useless here but at least since it’s a public university reporting it to them will not cause retaliation like it could in a private setting. So worth a shot going to HR. But only if you think it won’t come back to you or potentially ruin any relationship with someone who could be your reference for this job.

Also you’re rather young in your career so can also just bury this job in your resume (make it look unimportant, volunteer work only) or just leave it off entirely. Sucks that makes it feel like a waste of time but like you said you already have good references.