If you confirmed that the macOS Big Sur patch is requiring more space than what it needs to be installed in your equipment (or conversely, there is plenty of space on your HD but the software upgrade is insisting you're out of, or need more, space), the issue stems from a defect in the patch, and not from your equipment. Apple Support had me create a New User profile on my machine to sign in with and then attempt the install, which worked for me (and hopefully resolves the same issue for others as well):
1 Choose Apple menu > System Preferences, then click Users & Groups.
2 Click the lock , then enter an administrator name and password.
3 Click the add button (+) below the list of users.
4 Complete the fields shown for a Standard or Administrator account, then click Create Account (or OK).
...It is offensive to all technical disciplines (SDLC, QA and release management) for an enterprise like Apple to disburse a software upgrade--and one that is constructively mandatory, no less--on its customers that was clearly not ready for production. Guessing no one there ever heard of a Sev1 defect? This defect not only prevents the user from simply completely the install, which the user never asked for to begin with, but suggests the user's equipment is out of space (when it isn't), prevents unrelated files from being saved, and even solicits iCloud plan upgrades for otherwise unnecessary storage at a financial cost to the user! Anyone in Apple leadership who allowed such a deployment should be ashamed and take accountability (and take this feedback constructively: it is free, and you need to accept it, else it's time to move on to other career opportunities, since software development and leadership aren't your wheelhouse).