If you can download the Big Sur installer, then you can try creating a bootable macOS USB installer:
https://support.apple.com/en-us/HT201372
After creating the USB installer you can delete the "Install macOS Big Sur" app from the Applications folder and empty the Trash. Oh, that reminds me make sure you have emptied your Trash. You can also boot into Internet Recovery Mode using Command + Option + R to access the Big Sur online installer (again make sure to delete the app from the Applictions folder first to free up 12GB). I highly recommend having a macOS USB installer just to be safe even if you attempt Internet Recovery Mode.
You can also go into each of your web browser's privacy settings and delete their cache which can free up a lot of space sometimes. Also boot into Safe Mode which will also delete some temp & cache files from macOS. If you have the requisite 35GB of free storage space you should be able to install Big Sur from the USB installer.
Also make sure all of your backups have completed since the boot drive will retain snapshots of deleted data until the backups have all been transferred to the external drive(s). If your backups have all been completely transferred to the external drive(s), then you can check to see whether there are any remaining APFS snapshots that may contain the deleted data:
https://derflounder.wordpress.com/2018/04/07/reclaiming-drive-space-by-thinning-apple-file-system-snapshot-backups/
Before attempting the upgrade to Big Sur make sure to have a good backup just in case something goes wrong during the upgrade or if you find that you have issues (or dislike) Big Sur you can go back to your current setup with the older OS.
If these suggestions by @babowa & myself do not help, then you will need to contact an AASP to see if they will replace your drive with a larger third party drive assuming your Mac has a replaceable drive (not all AASPs will do this). If you have a laptop with a replaceable/upgradeable drive, then it is fairly simple to perform the upgrade yourself by getting the replacement SSD from OWC where they include the necessary tools and instructions (including videos).
FYI, you only provided us with the version of macOS used and not the exact model of the laptop. If "About this Mac" doesn't show the model, then enter your serial number here:
https://checkcoverage.apple.com/