There's no realistic way to upgrade to an intermediate version of iOS - you get to choose the version Apple offers when you do over the air (OTA) updates.
The best solution when an OTA update doesn't work is to connect to a computer so the device doesn't have to run the update itself and the only thing that can stop you from a good update is no space on the device or the device needing to be erased (which will then reveal if the issue was software failure or hardware failure).
In all cases - the runbook for this is:
- Review your backups - make new ones ideally to the cloud and to a computer if you have any data you care to retain.
- Power off the device - then plug it in to power and let it start up and charge.
- Consider changing networks to then attempt the OTA update.
- Delete any apps you don't care about, then review free space on the device.
- Consider offloading apps (which leaves the data you care about) but frees even more space to handle the update.
- Power off and then power on - try the OTA update.
- Take the device to a computer to try an update (this needs less space on device since the computer downloads and runs the install).
- Erase the device and then attempt OTA / computer assisted update with no data on the device.
- Restore your backup or start anew as you wish once the device is updated.
Apple support is also very useful in this situation - no matter which of the steps you are at - asking them for guidance may be worth the delay if you aren't sure on anything or run into a road block I didn't explain well enough.
On your side questions - you can run the feedback app on iPadOS to gather diagnostic logs or connect to a computer to get diagnostic logs. They are often only useful to engineering types or Apple, but you can get at the logs and find out what happened.
If you update from a computer - there will be different logs on that device as well. In your case, since you used a computer to update - it may be impossible to look back and understand what your block was, but next time you'll know and this might help others that face the same challenge as you.