I don't know anything about the Tailbeacon except what I've read online in the last 5 minutes, but it looks to me like it's only an ADSB-out device, and it also looks to me like the only time you need to connect to its wi-fi is during the initial setup process (to turn on the transmit function, set your tail number and other plane details, and set what ADSB-in frequencies you receive). That should be a one-time setup that's done on the ground, and then you don't ever have to connect to that wifi ever again (unless you ever need to change or repeat the initial setup for some reason).
So, have you properly done that, and confirmed that the settings "stick"?
If "yes", then it should just be doing its own thing and you shouldn't have to connect to its wifi anymore. So go fly, connect your iFly device to the Stratux, and that's it.
From your description it sounds like that initial setup has not yet been properly performed.