Byrd is generally regarded as one of the best FS in the league. He's only 26 years old.
Here are some reasons this might not be a good idea:
1) Goldson signed with Bucs for 5 years / $41.25 mil with $22 mil guaranteed. Byrd would demand a bit more than that.
2) A trade would cost some decent player(s) or draft pick(s).
3) By trading without a long term deal already in place, you place yourself at a disadvantage in the negotiation.
4) With Burnett's new deal, that would be too much money sunk into the safety position.
There were several quality, older safeties available in FA who signed for not a lot of money or years. Snooze...lose.