You revealed the answer in your question.
It's not hard to uncover where Superman lives... and you don't need an overwrought investigation or a linguist or drones. If push comes to shove, they can always find out from Lois Lane (whether directly or otherwise). This is the same government that seized her in derogation of her civil rights and who has a Suicide Squad program. Getting the information from Lois Lane is exactly what an unsavory government could do... and Superman knows this.
Instead, however, in this case the government took the high road.
Despite holding Lois they didn't subject her to "enhanced interrogation" and didn't disappear her (at least not beyond Superman's senses) or threaten her... and although they tasked a drone, that's "fair play." Based on the first article Lois leaked, it's clear that Superman didn't give her his identity at first, but the fact that she hasn't been reduced to a pile of ashes or sent one a one-way trip to the Sun indicates that Superman doesn't disclose his identity, but he also doesn't seem to mind if you find out. So the government endeavored to find out by legitimate means but Superman downs the drone and says:
"I know you're trying to find out where I hang my cape. You won't."
Note, he says WON'T not CAN'T. Essentially, Superman is saying, "You're going to stop investigating my identity because its one of the terms of how I'm going to help and you're going to agree to those terms." And, Swanwick does agree by conceding the point and moving on, "Then I'll ask you the obvious question...." If Swanwick didn't agree either that they wouldn't or couldn't determine his identity, he wouldn't concede by saying "Then...."
So why does Superman volunteer Kansas in the face of that? Well, why does he give Batman Kryptonite? Is it because Batman can't get Kryptonite on his own? No. Is it because Batman is the best person to use the Kryptonite against him? Not really, no, the advantage would be more useful in the hands of someone like Wonder Woman or The Flash. Is the Kryptonite the sum total of everything necessary to defeat Superman? Again, no. So why give Batman Kryptonite? Why give Swanwick a morsel of info on his identity?
It's a symbol that says, "Here's something that could lead to a vulnerability, but as a sign of my respect for you and trust in you, I'm giving you this... please extend me back some trust in return." How does Superman end the exchange?
"I don't know General. Guess I'll just have to trust you."
What did Superman ask for in that encounter? Swanwick to talk to Washington on his behalf, because a super-powered alien descending on the seat-of-power uninvited is terrifying and could cause a panic... how does our Batman v. Superman trailer open? Washington having invited Superman to the Capital Building. Seems like it worked and, for all we know, Swanwick kept that detail to himself.
Adapted from: http://www.manofsteelanswers.com/what-was-the-point-of-the-drone-scene/