Yes, you can grant any license you want to your larger work.
With respect to Creative Commons, they provide guidance:
May I apply a CC license to my work if it incorporates material used
under fair use or another exception or limitation to copyright?
Yes, but it is important to prominently mark any third party material
you incorporate into your work so reusers do not think the CC license
applies to that material. The CC license only applies to the rights
you have in the work. For example, if your CC-licensed slide deck
includes a Flickr image you are using pursuant to fair use, make sure
to identify that image as not being subject to the CC license. For
more information about incorporating work owned by others, see our
page about marking third party content. Read more considerations for
licensors here.
With respect to MIT License for software, I don't think that many of the reasons for fair use apply to using someone else's source code in your project.
If you're creating criticism, commentary, news, or educational material, you probably have more than just code. You should choose a more appropriate license for the complete work (like a Creative Commons License).
Taking someone's software source code and trying to use it under fair use may also lead to issues when you consider other factors, such as the purpose of the use, the amount included in the larger work, and the effect of value on the copyrighted work.
I'm not finding a lot of cases regarding fair use in software. Galoob v. Nintendo found that you can modify copyrighted software for personal use (not relevant to this discussion). Sega v. Accolade found that copying software for reverse engineering was fair use under certain conditions (again, not relevant here).
If you are attempting to use anything under fair use, regardless of the license that you apply to your larger work, you do need to ensure that you do not give the impression that each individual piece of that work is also under that licence once extracted. That's why you need to clearly mark which portions are used pursuant to fair use. If those portions are extracted from the larger work, then the original restrictions to use apply. However, someone can use the larger work under the license you grant.
I just wanted to add this brief section to be extremely clear.
When you are producing a work, you can choose a license for that work. If you are incorporating someone else's work into your own work, there are two possibilities:
You obtain the other person's work under a license. You must abide by this license and all of its requirements. Some licenses are viral in nature, which restrict the licenses that you can apply to the larger work.
You use the other person's work under fair use. In this case, you need to properly attribute the work and identify that it is not available under the same license as the larger work. Someone that extracts that smaller portion must abide by the copyright of that work. If it's available under a particular license, they can choose to use that license or under fair use if they are able to. Someone using your complete combined work uses your license.
