The manager asks for animated Splash screen, but I am against it, since the animation can slow down and annoy the user, but what do you think?
When in doubt, do user testing! Prepare both versions, at least in mock up, and let the users run through it multiple times. Since part of the concern is that users will be annoyed having to watch the animation every time the app loads, you can give them multiple tasks and ask them to exit the app in between. If the users are annoyed, then you have data to show your manager. And if the users don't mind or like the animation, then you've learned something.
As others have pointed out in the comments, a lot depends on when the app would look like without the animation. A simple animation may not actually have a noticeable affect on load time, and if the app needs to load a lot of data for core functions, the animation can actually give a cue that something is going on in the background. If the animation takes longer than the load time of essential functions, then you're wasting the user's time for no tangible benefit.