Our second son is now 8 weeks old. We chose his name months ago so it’s felt like his name for a long time. The name was traditionallly a nickname and when we told both sets of grandparents shortly before he was born they all felt that he should have the traditional long form of the name as his ‘proper’ name. At the time we thought that seemed reasonable so that’s what went on his birth certificate. However, I now massively regret it. We never use his full name but because it’s on his birth certificate it has been picked up by the doctor and appeared on a couple of official letters etc. We only put his everyday name on Facebook etc but still got cards that used the full version. I don’t like it and I feel that we have inadvertently inflicted a lifetime of confusion on our boy. I now really wish we had just chosen something else but it’s too late to change it now.
8 weeks is definitely not too late to change it if that’s what you really want!
You’ve already been exclusively referring to him by the nickname version of his name, so all it would take would be to update details at the Drs, and for everyone else a gentle “not X please — we just call him Y” if they use the other name. You don’t even have to say you’ve changed it officially, if you find that too awkward (which you shouldn’t!) At 8 weeks, your son will have absolutely no idea what’s going on; at that age mine were mostly called Pickle and [name_u]Pepper[/name_u] and Pops and all manner of other random pet names, so please don’t worry!
For what it’s worth, my son has a nickname on the BC and it’s never been an issue. Admittedly nicknames as full names are popular over here in the UK, but my son’s isn’t a particularly common one and still people are fine with it.
That said, it’s extremely normal to have a full formal name + “known by” nickname, so I really don’t think you need to worry about confusing your son even if you don’t change it. My husband is a [name_m]Maximilian[/name_m] almost always called [name_u]Max[/name_u], and he enjoys having that weighty full name at his disposal if circumstances demand!
Good luck.
If it’s going to bother you then it needs to be changed.
A birth certificate name is a legal thing, not necessarily a practical thing. No one has to know his legal name unless you want them to (or at the doctor, or something like that, but even so, you can tell the office "his legal name [name_m]Nicholas[/name_m], but he goes by [name_u]Nico[/name_u].)
I grew up with a legal name that was slightly different than what I was called, and it was fine.
If anything, this will give him opportunity when he gets older to go by a more formal moniker, should he ever want to.
I mostly agree. I’ve gone by a nickname from a longer name my whole life, and it’s never really bothered I just say at front offices etc that “I go by X not Y”
At 8 weeks you should be fine to change it if you want it to be. and yes, if he chooses to go by his longer name for any reason he still has the ability to 
If you don’t really like the longer version on the bc I think you should change it. It’s not a big deal to change it and I don’t think you need to feel any embarrassment about it. While I also agree with the pp ideas expressed above about comfortably living with 2 versions of their name, this only applies if you as the parent actually like the name on the bc, which you don’t seem to. Both changing and not changing the name are fine, but what would make you happier in the end?