Why can't I dress up my sim in this costume?

The chicken costume is a career outfit. You have to unlock it through the athlete career.

There are several outfits that should have been included in the choices for costumes, but for whatever reason or oversite EA didn't include them.
There are though mods that do address this issue. Mod the sims has one, but it hasn't been updated since August, but they do let you know there is a better version over at another location. Here is a link to that. http://modthesims.info/download.php?t=614780

From what I can find over at that location is this.

https://deichschafblog.de/bienchen/sonstige-mods-misc-mods/ Look for the one called partytoddlers
https://deichschafblog.de/bienchen/grossstadtleben-city-living/ Look for the one called nerdfestival_version
https://deichschafblog.de/bienchen/werde-beruehmt-get-famous/ Look for the one called wearcostumetweak

There is also another over at this location.
https://littlemssam.tumblr.com/post/175412396783/random-small-mods-sometimes-mods-are-sooo-small
To read more about it, it will be in the section called Seasons DLC

Both these people do a pretty good job in keeping all their mods updated when new patches are released.

Thanks @Gagea and @LaBlue0314! @Gagea, that explains why it was available to restaurant staff and not to my sim (who is not in the athletic career). Will take a look at the mods @LaBlue0314 !

That’s a llama, not a chicken.

A llamacorn, to be precise. It's not yet available in my wearcostumetweak but I have added it for you:
http://bienchen83.bplaced.net/s4/mods/sims4me_wearcostumetweak.rar

Thank you @83bienchen!