Due to Lex’s Africa move, there’s now an official government-sanctioned narrative of accountability. Lex can pitch his silver bullet deterrence plan (and ask for ship access as a part of that).
If his import license is granted, that means Senator Finch is on his wavelength. She sees the need to keep the monsters in check. She supports his public deterrence of Superman which puts him in his place and sends a message to all the hidden metahumans to stay in theirs. Lex is okay with this outcome. It satisfies his motive.
If the United States Government is above Superman, holds him accountable, and finds having a weapon of assassination against him as acceptable, then he’s won. People see Superman as the threat that he is, Superman is subjugated and not “you above all” and there is no more existential crisis. Lex can simply stop, his plans and machinations and moves can stop here. On the off-chance Superman starts to win back the people and the sentiment shifts, well, Lex can always steal the Kryptonite from himself and use it to kill Superman first. Of course, that’s not what happens, Senator Finch blocks the import license later on.
In the meantime, under the narrative Lex created, it’s reasonable to ask for access to Kryptonian artifacts- the ship and Zod- to research and develop alternatives to Rocky.