iMac locked and bluetooth keyboard won't connect

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. The bluetooth mouse is connected and working, so the iMac bluetooth appears to still be working. I can also pair the bluetooth keyboard with another device (e.g., my ipad, iphone), so it also appears to still be working. The keyboard simply stopped pairing with my iMac, but because I can't log in, I can't further trouble-shoot. (After confirming the keyboard still works by pairing it with my ipad and iphone, I disconnected and forced those devices to "forget" the keyboard, so there should be no conflict. I have also rebooted both the iMac and the keyboard several times.)

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. Any suggestions?

Thanks