does "early Bird" rights mean we can be over the cap and still sign him without his salary being counted against any of the exceptions?
Yes. If BBD signs a deal for the MLE or less with another team, we can resign him and preserve all of our other exceptions.
The only part I'm unsure of is this: Other teams can offer BBD a deal where the first two years are the MLE amount, but in the third year the salary jumps drastically. If that happens (which it won't) I'm not sure if we could match strictly using early Bird rights, or whether we'd instead have to give up our MLE. Larry Coon is ambiguous on this, and I haven't gone back to the CBA to look into it further.
But, long story short, so long as BBD doesn't get an offer for greater than the MLE (in year three and beyond), we'll be able to match and still use the MLE.