I see two possibilities here:
The Simple Argument:
The key line on the Symbiotic Orb is "This counts as a Defense". So, despite the wording of 'once per round when an object in your zone is melee attacked...' the card should activate (and be paid for) at the appropriate time in the attack sequence for Defenses, which is at the start of step four: Avoid Attack.
Now, during attacks, Enchantments can only be revealed at the end of any step, not during one.
So the simple reading is that if the player with the Falcon Precision wants to use it against someone who might be defended by a Symbiotic Orb, the FP needs to be revealed no later than the end of the Roll to Miss step, because the Orb would be activated during the Avoid Attack step, before the FP has another opportunity to reveal. You thus could not drain 3 mana by revealing the FP after the Orb is used.
This has certainly been the way that my group has used the card so far.
The Counter Argument:
The key line is "when an object in your zone is melee attacked..." It does not say "during an attack." So the 3 mana needs to be paid at that time the attack begins, at Step One: Declare Attack, and is paid to stake in advance a claim that a Defense will be used at Step Four: Avoid Attack.
If this reading is right, then the Falcon Precision could definitely be used as a de facto 3 mana theft, and the forcemaster holding the Orb needs to think a bit more about whether she wants to use the Orb. ("Oh, the attacker has a Daze condition, so the attack might not get through anyway... Do I risk it?")
I can certainly see the appeal in this argument, since it does balance out the very powerful automatic Defense provided by such a cheap piece of equipment.
Does anyone more knowledgeable of the card's intent have anything to say?