Hi James! I hope you are well. My name is, Zadee. I'll try my best to help you today.
From what I understand, you can pair a BT device to multiple devices but you may not be able to use it on those devices at the same time. This is just like the BT earbuds scenario I had the other day. My earbuds are connected to multiple devices, including my laptop, android devices, and others. I sometimes forget to turn off the BT on my laptop, so when I turn the BT ON in my android device, the BT connection would establish but I won't hear anything from the android because it auto-connects itself to the laptop. So I would either have to turn off the BT on my laptop or re-pair the BT earbuds to my android. That said, yes, you would have to re-pair that keyboard to whichever device you want that connected to.
Have an awesome day/evening!