To close out this thread for future reference, it turns out that my anti-malware software, Malwarebytes, was preventing the install from reaching the auth server. Turning it off temporarily allowed for successful installation. Also, installing from the zip files worked as well, as long as I made sure my github repo was private so as not to expose the club files.
I'd like to give a huge shoutout to Rodrigo for amazing support. He was super proactive in helping me through the troubleshooting process over a period of several days. It was clear he really cared that I found a solution. I feel like GSAP's community support is way better than a lot of the paid support one gets with other types of software. Love it.
I'd also like to thank Prasanna for going above and beyond to help me make sure it wasn't just some simple mistake I was making. Thank you both!
TimTh's post in Wrong version error on club install with npm was marked as the answer