Even when they’re on the starbase at the start? I figured that was the real Mendez - only the one from the shuttle onwards was the illusion maybe?
Maybe, but then, at what point could the illusion have been substituted for the reality, if Kirk and "Mendez" were together the whole time from the starbase to the shuttle? The answer is of course at any point. We can never be sure, much as we cannot with respect to virtually anything presented to us in this story, from beginning to end. Just how deep does the rabbit hole go? No way of knowing. Was Spock indeed lying about receiving a message from the starbase, or did he rather receive an illusory one? Was Pike even ever injured at all, or was this all the Talosians' second—and this time quite successful—attempt to entice him into their menagerie with an elaborate ruse? If their powers are as potent and far-reaching as is alleged, then they are certainly capable of it. And it would make a fair bit more sense as a motivation for their purported machinations here than it all being simply out of the goodness of their shriveled hearts! Ultimately, as @Timo pointed out in a previous discussion: "Whatever the Talosians did, they won. By definition. If we thought otherwise, that's only because the Talosians wanted us to." -MMoM
Everything in Star Trek since that episode has been a Talosian illusion, except whatever took place before it.
Don't forget he originally was a voice for the lead Talosian in TOS - "The Cage" (His part was re-dubbed when they had him play Commodore Mendez in TOS - "The Menagerie")... So, he's not just a double agent - he's a TRIPLE agent (and probably a Talosian projecting mental images of himself as a Human, Romulan and who knows what else!)
You're right, that's totally less believable than the thread's initial premise. -MMoM [EDITED TO ADD: No offense intended, @Groppler Zorn!]