IANAL, but the case you describe is IMHO similar to the distinction between licenses for individual modules of a software and the software "as a whole", as you often find them in open source software. For example, though the Linux kernel is licensed under GPL, it contains parts or modules which have a different (compatible) license (like libc, which is under LGPL). You will typically find different license texts in separate files for such a case.
The MIT license reads "... The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. ...", but it does not contain any requirement to put the copyright text into a file named "LICENSE.md".
Putting this together leads to
you have to add the original, unmodified copyright notice from Clementine.js somewhere to your programs docs (so this rules out any kind of "merge" where the original text is modified)
you should have a clear license distinction between the final program "as a whole" and the license of "Clementine.js"
the license you use for the final program must be "compatible" with the MIT license (it must not contain anything which contradicts the licenses of any of its parts or "foundations")
If you achieve this by creating two separate license files, or by writing both license texts into one file, does not really matter, there is nothing in the MIT license saying the text must be kept in a separate file. Mentioning in your license text that the final product was built off of Clementine.js is a good idea for both cases, independently from the decision for one or two files.