Unable to optimize iCloud storage or update iPhone due to low storage

I am having trouble getting access to storage optimization with my new iCloud subscription. I worry that I'm stuck in a catch-22 between not having the update necessary to clear up storage, and not having enough storage for an update.

I had a baby recently and the explosion of photos has taken a lot of storage. I got an iCloud subscription so I could optimize storage and get some relief on the tight storage. But whenever I try to back up my iPhone (which I understand is a necessary preliminary step to the storage optimization) the back up goes on indefinitely and the estimated time only gets longer.

I've read online that it could be because I need to update my phone. And my phone is due for an update. But I don't have enough storage for the update. It feels like a catch-22. Wondering what solutions might be available?

Thanks!