Did you power cycle bluetooth? Also does the nexus 7 have bluetooth 4?
The original nexus 7 only has bluetooth 3 https://www.gsmarena.com/asus_google_nexus_7-4850.php
The 2013 version does have BT 4, but honestly, it's a 6 year old tablet - I'm not too surprised it doesn't work well.
That is gen 1. Gen 2 have more of a straight edge not rounded and have an etched line around edge not smooth.
See photo of gen 2 here. https://flic.io/shop/flic2-starter-kit
Gen 2 are also only white so your photo was white but if you see colored ones they are gen 1.
Can’t help with where to get in Australia but perhaps directly from flic.
I hope some of the talented people here in the community can help by making a 3D printed design to make the button bigger and easier to press.
But for the meantime my advice would be to get the original flic button (Link: https://www.amazon.com/Flic-Wireless-Smart-Button-Black/dp/B00ZAENDSC/) Because the original button were convex in shape and much easier to press with a foot for example, on the contrary to the new design which is concave and requires a more precise press!