You, the player, do not have the option to Target Lock Kagi under these circumstances. You are not getting a free target lock action ala Dutch Vander, that would allow you to pick a target in range. You are being handed a target lock via a different mechanism that can only apply to the target you shot at.
This is much less about Kagi and much more about Fire Control. Nothing on Kagi's text would allow him to reduce or eliminate any restrictions your ship may have. There is no text on Kagi that allows your ship to target lock any ship other than the one that you shot at. In fact, the only text that refers to the entire situation is the part where Kagi only requires you to TL him "if able".
You are not able. There is no possibility for you to gain a TL vs. Kagi using a Fire Control System (unless you shot at him, of course). So it seems to me this situation is actually pretty cut and dried. Kagi says if able, you are not able, so no need to TL him.
Hypothetically, lets say there were a weapon that would allow you to target multiple targets. If you were to fire at Kagi and Darth Vader using a Fire Control System, then Kagi's ability would require you to put the TL on him.
Edited by KineticOperator