This is fairly easy and straightforward though it requires more than one Apple device. Ensure that you’ve got handoff/continuity turned on for the shared clipboard on your devices.
Install an app like PastePal or another similar app that has iCloud sync and the ability to automatically monitor the clipboard. PastePal is a one-time payment and works on iPad, iPhone and Mac. My iPad Mini is always within reach as is my iPhone. For now I’ve added the app to both of those device home screens. It does require that the app be active on the device, not in the background. So it’s not a perfect solution. In theory a nearby Mac should also work but my Mac often stops seeing shared clipboards whereas the iPhone/iPads almost always work.
Anytime I copy on my iPad Pro my secondary “PastePal Server” gets the clipboard and it syncs back to my iPad Pro almost immediately. I’ve been using it all day and it’s worked flawlessly. Editing to add: It’s important to check the settings for the app. Turn on Monitor Clipboard, iCloud sync as well as “Save images in iCloud Drive”.
I’d tried this awhile back using an app called Yoink which also has iCloud sync and clipboard monitoring. It should work in the same way but I found it unreliable so gave up on it. Yoink does okay to monitor the clipboard on the iPad using a kind of hacky method using the picture-in-picture to remain active. But at some point fairly recently I noticed that not all clipboard items were being reliably saved in the app.