How do I update my software without making any drastic changes to my iPad?

You appear to have a dilemma. If understood correctly…

• Your iPad storage is so full that you have insufficient space to download and install an OTA system update directly from settings.

• You are unwilling or unable to use Cloud Storage (such as iCloud) - with which to move data.

• You either don’t have access to, or are unwilling to use, a PC or Mac computer (recognising that update by a PC/Mac is possible where locally available space is constrained).

As such, you have eliminated all potential paths to complete the update. A base-model 16GB device is always going to struggle with available space if you store data locally.

The only remedy to having too little space on your device is to identify and move/remove [unwanted] data. You may find these help pages offer some useful information and insight:

How to check the storage on your iPhone, iPad and iPod touch – Apple Support

About storage on your device and in iCloud – Apple Support

If, when looking at your own data storage usage, you see that the “other” category is consuming an disproportionate amount of space on your iPad, this support page may be helpful to you:

Clear Other storage on your iPhone or iPad - Apple Support

Consider that iCloud storage is not expensive. You can upgrade to an adequate “paid tier” as needed, downgrading again (back to the free 5GB quota) when you’re ready to do so. If you don’t have access to a PC or Mac, I suggest that you either use an iCloud backup - or manually move data to iCloud storage - prior to attempting the OTA update.

This iPad is woefully under-configured for your current usage and your no-connections preferences, and it’s only going to become more storage-constrained as you upgrade apps and iPadOS.

In addition to the above, if there is no computer and no iCloud usage, there are no complete or no automated backups. iPadOS doesn’t (yet?) support direct local backups. Which means this iPad is one failure or one theft or one loss or one corruption away from catastrophe. Manually-managed backups are possible, but that’s not going to be easily restorable.