You guys have to much faith in the cat. And are forgetting it's Shinjiro. Yes, out of all of them, Shinjiro is the most trustworthy. The problem however, is that Shinjiro has mental trauma himself that constantly acts up whenever we're around. Now, in reply to that you're probably going to say something like "This shows we trust him" or "This shows we like him" or something like that. The thing is, that's the problem. There's a reason Shinjiro is in therapy. Ken liking or trusting him? That just makes his constant guilt worse. It pours salt on that wound he has. Hating him isn't any better, but the implications of telling him is not going to go well with him.
And it doesn't matter that we tell the cat to avoid Ikutsuki. He'll most likely go after him. Remember, this is the same cat that runs away when he realizes he's not the best thing since slice bread to someone he can convince he's the best thing since sliced bread. He only "listened" to us because his cocky, self absorbed ego thought that he'd be able to convince us to go to the Shadow Nest after showing off his strength. And I do mean cocky. Ken, who doesn't have much experience using his powers, can tell when he's getting low on SP. But the cat either couldn't or ignored that he did to appear great. He'll go after Ikutsuki because he assumes he can, despite all evidence saying otherwise.
Also, assuming by some miracle he doesn't go after Ikutsuki, the risk of him blabbing about being a time traveler is still to great to me. Again, citing his ego that makes him desire heavily to be special in the eyes of others. Even if he doesn't tell everyone about our own time traveling, he'll brag about his own. Brag about the Phantom Thieves. Hell, I'd need to check, but I'm pretty sure he bragged about being part of the Phantom Thieves of Hearts before even learning we were also a time traveler. And even Shinjiro will tell at least Mitsuru about Morgana because Shinjiro can be responsible at times and a time traveling cat is definitely something at least Mitsuru should be aware of in his mind. Especially since Morgana can also sense Shadows.
So no. Bringing the cat to anyone in SEES is a bad idea.