Big Sur Disc space issues...32GB free space insufficient

I can't be certain. In addition to their usual creation provided "Back Up Automatically" is chosen, Apple explains Local Snapshots are recorded prior to every macOS update, a fact I personally confirmed. Refer to About Time Machine local snapshots on Mac - Apple Support. They are very useful for that purpose. But I can find no reference to their applicability to macOS upgrades, and given the extent of this one I seriously doubt they are.

Another option you might consider is upgrading that Mac's 128 GB storage module with an aftermarket one of larger capacity. If you want to explore that option the only vendor I recommend is OWC / MacSales. Their reputation is unmatched.

128 GB was never generous to begin with, but it just isn't enough to be useful any more. Literally every complaint about running out of space comes from MacBook Air users with 128 GB storage like yours. Personally I think replacing it with a larger capacity storage module is the only option worth considering if you want to keep that Mac in service.

Just be advised that it is not considered an "authorized" modification, meaning that Apple will henceforth refuse to service that Mac at any cost. If you are still using its original battery, seriously consider having Apple replace it before it becomes "vintage", and definitely prior to altering its hardware.

Excerpted from How to upgrade to macOS Big Sur - Apple Support:

"If upgrading from macOS Sierra or later, macOS Big Sur requires 35.5GB of available storage to upgrade."

Upgrading from macOS versions earlier than Sierra will require 44.5 GB.

By the way I just realized I assumed your MBA only has 128 GB storage 😆

You never said it did. But I'll bet my assumption is correct.

By the way I realize I did not directly answer your question, which as I understand it is how much "available storage" will remain after the upgrade. The reason is that it's a moot point if you can't reclaim enough space to even accomplish the upgrade. I don't have a definitive answer to that question, but I can suggest a workaround.

Obtain an external storage device such as a USB hard disk drive. It doesn't have to be particularly large fast or expensive but if that's not a concern the best choice would be one with a Thunderbolt port (the older Thunderbolt, not the newer Thunderbolt which is more commonly known as USB-C). $50 will buy one that's acceptable for the purpose.

Boot macOS Recovery, format that disk, and install macOS Big Sur on it. When it finishes, it will ask if you want to transfer content from another Mac. Do that, the "other Mac" in this case being your MBA.

When it finishes you can boot from and use that external hard disk drive as if it were an internal one. The only difference is that it will work a lot slower than the MBA's internal flash storage, primarily due to the slow USB connection speed but also a slow, rotating hard disk drive. (An external SSD with a Thunderbolt port would be ideal because it would be unaffected by either one of those bottlenecks. It's just expensive.)

At the end of that exercise you'll have your answer, and you won't have to alter your MBA itself in any way whatsoever.

Finally, when you decide you can use Big Sur to your satisfaction and that the resulting installation conveys sufficient excess "available storage" to keep using your MBA without alteration, you can repeat that exercise with the MBA's internal storage. The external hard disk can then be pressed into service as a Time Machine backup drive—which you ought to have anyway.

This essentially describes the technique I use whenever upgrading macOS on any Mac, regardless of its storage limitations or other concerns. It's a very conservative approach.

The actual installed OS for Big Sur is only a couple of GB larger than Catalina. But one feature (if you want to call it that) is Big Sur makes and retains a snapshot of the system prior to the update, which is about 8 GB. That snapshot is retained until the next system update.

So, overall a Big Sur install will take about 10 GB more than a Catalina install. You could avoid the snapshot by doing a clean install, but that would mean you’d need to repeat that clean install with every incremental update to Big Sur.