It should not be necessary to pair/unpair the keyboard each time. If the keyboard has an on/off switch (and it should) then turn it off when you want to use the iPad’s touch keyboard. It should re-pair within a few seconds when you turn the keyboard back on.
Some third party keyboards made specifically for the iPad have a keyboard key. If yours does you can raise and lower the touch screen keyboard using that. The key usually has a keyboard icon.
You can use the globe key on some hardware keyboards to trick third party keyboards to appear. Tap the globe key and select the Emoji keyboard. It will come up. Tap the globe key again and select a third party keyboard. It should appear. If you select the default iPad keyboard, the screen keyboard will disappear again.
Of course you have to have both the Emoji and a third party keyboard enabled for this to work.
Also, I’ve only tested this with Apple’s smart keyboard/case for the iPad 12.9. It might be fluke, and useless on other keyboards. It’s awkward, and possibly a bug rather than a feature.