I'd say you should do the FA-10 even if the shop did one. The lower doesn't meet the criteria to fire a round. When you build it, it does and that's when it should be registered.
If having two entered was an issue, everyone who built a lower from places like Four Seasons would be getting calls.
Not really. Maybe they are smart enough NOT to re-invent the wheel and just use the gun that was already registered (even if improperly).