Hi,
I was reading this topic and was about to return the keyboard, as I often need to pair the keyboard between my machines (more than 3, so I register the most used ones depending on my current usage). I had the exact same LED behavior you described: slow blink in the beginning, and then the 3 fast blinking sequences in a loop. Spamming the Bluetooth key where the LEDs are did *nothing* and the LED animation continued to play in a loop.
But I found a solution!
The problem was that when the keyboard was last used, I had disabled ⌨ FnLock on the keyboard! So, instead of pressing the ⌨ Bluetooth key, I was effectively spamming the ⌨ F1 key and the keyboard was just trying to connect to send the key to the previously paired device.
The solution was to press ⌨ Fn + ⌨ Bluetooth (instead of ⌨ Bluetooth) in order to switch to the next pairing device, and I was able to pair my other computer! I guess trying to disable FnLock would also work, but I do not remember how to do that so I just use Fn+Bluetooth instead.
Hope this helps. 🕵️♀️