Quick update to this old thread about the Gecko tag: I was invited to join their beta app program, so I accepted. The beta version I had installed (2.1.4) now pairs the tag as a proper Bluetooth device (security code 0000000), and as such it can now be used as a trusted device. I tested it and it works great, even immediately after a reboot without having to first launch the app (the usual requirement to enter your PIN once still applies).
My device (Moto X 2013) appears to have horribly broken Bluetooth, however--when I have it turned on it kills the battery and severely interferes with WiFi communication--slows transfer speeds to a crawl, frequent dropouts, etc. Although now I'm wondering if it's possible these ill effects were caused by the Gecko app. I just installed the latest beta (2.1.5) and at first glance WiFi is still working well. I'll leave Bluetooth enabled tomorrow and see how it goes. Crossing my fingers...