Well, at least a short time ago, you didn't need the extra badges to create the beacons - I had a group that didn't have the PI beacon, for instance, and I just needed the original one (out, right from the ferry to the - second? - island) to get it.
Do they show up in SG badges? I'm assuming not. (I'll have to check one of mine that hasn't changed for a while.)
Edit: Checked two of mine that haven't changed in quite some time, all the bacons are still there and available in the edit window.
Nevermind, they showed up finally about 20-30 minutes after I upgraded my base plot.
Which is good, I would have been pissed if I had to get every single beacon again.
So I'm redesigning my base(its been about three years) and I cannot find any of the base teleport beacons. I have previously gained them all.
I have a guess as to what happened. I acquired all the beacons before the extra exploration badges were added to the game. Does this play a factor?