I spent hours scouting the internet for a solution to my problem, but I eventually was able to get through the backup of my chats in the WhatsApp Business app.
Since this is new iPhone season, I assume a lot of WhatsApp Business users will be in the same boat as me, so I’m hoping that posting the solution here will help others.

WhatsApp backup to iCloud stuck at 99%
No matter how many times I tried, my backup to iCloud was getting stuck at “preparing 99%“. It eventually would throw the following error message: “Couldn’t complete backup. Tap “Back Up Now” to try again. Keep WhatsApp open until it starts uploading.”
Thinking it was a storage space issue, I went into WhatsApp Business > Settings > Storage and Data > Manage Storage > Chats, and deleted all images and videos from my largest chats. That helped regain several GB of storage on the device. But still, I would get stuck at 99% despite force closing the app, and rebooting the device multiple times.
The solution
I tried every possible solution you can find on the internet (mostly on Reddit), including rebooting multiple times, force quitting the app, enabling and disabling end-to-end encrypted backups, but nothing helped.
I knew from the beginning that this last option would probably be the one, but I just didn’t want to do it because it has notable downsides.
So, reluctantly, I eventually logged out of iCloud on my iPhone and logged back in. You can do that in Settings > Your Account > Sign Out.
Immediately after doing that, I went back to back up my chats (which were now down to about 2GB of storage), and it worked. It took forever to prepare the backup, so did the actual backup of the data to iCloud, but it worked!
Pro tip: make sure you go to Settings > Display & Brightness > Auto-Lock > Never, to make sure your screen stays on and your device doesn’t lock during the process.
Restoring the WhatsApp Business backup to the new iPhone
On my new iPhone, I simply opened WhatsApp Business, verified my number, and then it automatically pulled up my recent backup. I hit the Restore button, and all the chats came back almost instantly. However, the photos in each chat took a while to come back.
To really fast forward the process, I really suggest staying on Wi-Fi, and leaving the WhatsApp app opened. Even then, the restore of media files seemed to get stuck at times, as if WhatsApp itself was timing out. In which case, I would force-close the app, open it again, and the restore of media files would start right back from where it left off.
The same issue happened with indexing of chats data. I use the Search feature of the app a lot, and all that data had to be re-indexed. Just like with the restore of media files, the indexing seemed to get stuck at times, so I simply force-quit the app and opened it again. It took a bit of time, but it worked.