By calling into the Flickity library from your project, you are creating a derived work as far as the GPL is concerned.
However, this does not mean that your code must be under the GPL license as well. Your code must be under a license that is compatible with the GPL (which the MIT license is) and people who make changes to your code have to comply to the rules of the GPL license.
This means effectively that it is not allowed to make a closed-source (derived) version of your project without removing the dependency on Flickity.
The only way to include the file and use the "open-source licensing" would be to change your license to GPLv3 (otherwise you need the commercial license).
Side note: changing your license to GPLv3 does not mean everyone that uses your product has to provide the sources for it (this would only be covered by the AGPL).
Therefore the main impact I see is: if someone changes your code and wants to distribute it he has to use the GPLv3 himself for the distributed source.