For the reason stated in your question ...probably
So far, we've not been given a direct explanation for why Lorca behaved this way, he could very well have just been behaving as Starfleet regulations require. But going by the events of the previous episode, and what we learn in later episodes (his treatment of Stamets when battling the Ship of the Dead), we could assume the following:
Lorca knows Cornwell is going to strip him of his ship after she returns from her mission. When she's taken hostage, Lorca see this as an opportunity, he gets to keep his ship and continue his war against the Klingons.
He's more than willing to sacrifice any single individual to achieve his goals. Stamets and Cornwell so far at least.