By definition, a firearm with an antique receiver is an "antique" and not a "firearm", so it does not have to be recorded on an FA-10.
As far as changing the barrel or other parts go, as long as the replacement is original equipment (even if made after 1898), the gun would still be C&R OK. The place where people get in trouble is when they try to mount components that were never original to the gun and try to sell them. For example, it is possible to fit a certain machine gun magazine onto an Egyptian Hakim to increase it's capacity from the standard 10 rounds. But, it is not "standard" equipment, and would void C&R qualifications.
The above is based on MY reading and understanding of the rules. For more definitive information, read the regs yourself, or check with your local BATF office.
*