Thinking about changing my babys name

So my son is now 3 months old. We proceeded to name him Benjamin Anthony because quite frankly we didn’t liked The other options.

We call him Benjamin and nickname is Benji.

But this whole time I feel as though his name doesn’t suit him! We do like his second name Anthony but I dont know if we should LEGALLY change it or leave it as it is and just call him by his second name?

Has anyone ever done a name change in this way?

Would love to hear some feedback!