iMac locked and bluetooth keyboard won't connect

2013 iMac and Bluetooth will not connect to the keyboard, so I cannot login.

Have tried unplugging , restarting several times but wireless keyboard will not connect.

I've confirmed that the keyboard is working properly after linking it to another mac.

Since the iMac is locked (I have password enabled on start-up and screen saver), I need to enter the password, but I can't because the bluetooth keyboard will not connect.

My next step is presumably to order a USB keyboard and use that to log in and trouble-shoot further, but I certainly don't want to waste money on a USB keyboard if I don't have to.

Is there any other option?