You've got it reversed there friend. Muslims would already go as far as openly denying Muhammad if it meant furthering their interests. Plus Jesus is still a holy person and a prophet in Islam, so they'd have less trouble saying it anyway.

But Jews reject Jesus completely. He's a false prophet in their eyes. It's them who he'd offend so much, because the whole idea of him is basically saying "Your covenant is invalid and you are not the chosen people".