Can I Pair an Apple Wireless Keyboard with multiple iOS devices (not simultaneously) without ‘forgetting’?
The keyboard does support been paired with multiple devices at the same time. However, there's no "switch" to toggle between devices - neither a physical switch nor a software switch in an app or similar.

The only way to make sure the device you want to use is the one currently communicating with the keyboard is to ensure that it is the only one within Bluetooth range.

If your phone stays in your pocket, it is not that practical to get that device out of the range of the keyboard. A "hack" is then to simply disable Bluetooth on the device that you do not want to use with the keyboard with currently. When you want to switch devices, disable Bluetooth on the current device, and enable it on the device that you want to switch to. The keyboard usually switches over in the matter of seconds.

If you wanted to use Bluetooth for something else on either device, the above hack won't work.

There's no other solutions through third party apps or similar.