Picard believes in effective responses to situations.
So the Romulans are constructing a secret base somewhere. It can either be confirmed under cloak, reported to Starfleet Intelligence, allowed to enter the same circuit of informational subterfuge which Romulans show time and again to have an extreme affinity and appreciation for...
Or, quite separately, Picard can leverage his good standing with the Klingon Empire -- a mutual adversary to the Romulans -- to display a willingness to investigate the plot out in the open, to stare Tomalok in the face and call out the attempted deceit, and when prompted to flex, do so with a bunch of angry Klingons in the background.
He's posturing, yes, but he's also showcasing that he doesn't have to play sneaky Romulan games to get after what he needs. One thing Romulans just seem to despise (and who doesn't -- but Romulans, more so) is being outwitted. It isn't enough to just gain the information; the information is a McGuffin, the spy game is paramount. It's the old 'you saw him, he had a gun' routine; they create a situation to which the Federation must respond, only to call them out for a treaty violation and respond with 'I was acting in self-defense', while giving the Romulan Empire full rights to declare war. Picard throws all that away; he takes the flagship of the Federation into the Neutral Zone, leveraging his violation of treaty against their own, and then his Klingon friends decloak and he sends Tomalok back to Romulus with his "u mad bro" face on.