Well, a week later and the problem persists. I guess I get to open up a ticket.

But here's the weirdest part; I decided I'd record it happening so that I had something to show Tech Support. But every time I entered *28 and the phone number, it worked! Twelve times at two different times of day, if I dialed *28877xxxxxxx and then the conference code it worked every time. I alternated that with just dialing 877xxxxxxx and the conference code to ensure I was still having the problem.

So why is it that dialing the number directly creates questionable tones that the conference line sporadically interprets as double tones, but dialing with *28 in front does not?

I guess I'll open a ticket and find out.