There are many Chapters where such mix of both Medinian and Meccan verses are to be found. Why is that so? Why would the compilers of the Qu'uran put a Medinian verse in that of a Meccan Chapter or vice-versa? Doesn't that imply that verses are being taken out of their chronological context to fit another criteria?
The order of the verses (ayāt, Arabic: آيات) within a chapter (sūrah, Arabic: سوره) is by Divine revelation through Jibrīl. When a verse was revealed to the Prophet ﷺ, Jibrīl told the Prophet ﷺ where the verse goes, and it could very well be a verse revealed in Medina to go after or before a verse revealed in Mecca in a surah that is considered Meccan.
For example, Chapter 16 (Sūrat Al-Nahl) is agreed by scholars that it was mostly revealed in Mecca. After the battle of Uhud, several of the Muslim corpses were mutilated by cutting off their ears and noses. They wanted to take a much worse revenge for the mutilation that took place. There and then, on the battlefield of Uhud, the 16:126 was revealed ordering the Muslims to punish only to the extent of the harm that took place, and to be patient is even better. This was documented in Musnad Ahmad in the following hadith (abridged below):
قال ثم شخص ببصره فقال أتاني جبريل عليه السلام فأمرني أن أضع هذه الآية بهذا الموضع من هذه السورة
— NOTE: My own translation, so treat with care:
He said that he [the Prophet] stared, then he said: "Jibrīl came to me and ordered me to put this verse in that place of the sūrah."
This shows that verses were revealed and ordered with a chapter of the Qur'an by Jibrīl. Furthermore, Jibrīl would review the entire Qur'an with the Prophet ﷺ once every year, and twice in the year in which he died, which was narrated in Sahih al-Bukhari (abridged below):
إِنَّ جِبْرِيلَ كَانَ يُعَارِضُنِي الْقُرْآنَ كُلَّ سَنَةٍ مَرَّةً، وَإِنَّهُ عَارَضَنِي الْعَامَ مَرَّتَيْنِ، وَلاَ أُرَاهُ إِلاَّ حَضَرَ أَجَلِي، وَإِنَّكِ أَوَّلُ أَهْلِ بَيْتِي لَحَاقًا بِي
The Prophet (ﷺ) said: "Every year Gabriel used to revise the Qur'an with me once only, but this year he has done so twice. I think this portends my death, and you will be the first of my family to follow me."
Having said that, even the companions themselves had questions similar to yours. Ibn 'Abbās asked 'Uthman ibn 'Affān for the reason for the why specific verses and chapters were ordered, and 'Uthman responded (abridged below):
كَانَ رَسُولُ اللَّهِ صلى الله عليه وسلم مِمَّا يَأْتِي عَلَيْهِ الزَّمَانُ وَهُوَ تَنْزِلُ عَلَيْهِ السُّوَرُ ذَوَاتُ الْعَدَدِ فَكَانَ إِذَا نَزَلَ عَلَيْهِ الشَّىْءُ دَعَا بَعْضَ مَنْ كَانَ يَكْتُبُ فَيَقُولُ ضَعُوا هَؤُلاَءِ الآيَاتِ فِي السُّورَةِ الَّتِي يُذْكَرُ فِيهَا كَذَا وَكَذَا وَإِذَا نَزَلَتْ عَلَيْهِ الآيَةُ فَيَقُولُ ضَعُوا هَذِهِ الآيَةَ فِي السُّورَةِ الَّتِي يُذْكَرُ فِيهَا كَذَا وَكَذَا
A long time might pass upon the Messenger of Allah (ﷺ) without anything being revealed to him, and then sometimes a Surah with numerous (Ayat) might be revealed. So when something was revealed, he would call for someone who could write, and say: "Put these Ayat in the Surah which mentions this and that in it."
Accordingly, the scholarly opinion is that which was documented by As-Suyuti in Al-Itqān 1/214-215 that Al-Baqillani said the order of verses and chapters was given by Jibrīl, as was the opinion of Imam Mālik, Al-Baghawi, and others.
A chapter being Meccan or Medini means it began to reveal before or after Hijrah.
It could include verses from both times since not all of the Surahs were completed at once and instead were revealed in stages, depending on the situation that needed to be addressed.
The Quran was arranged by Allah Himself during the Prophetic mission:
“And those who disbelieve say: Why is the Quran not revealed to him all at once? [It is revealed] thus that We may strengthen thy heart there with; and We have arranged it in right order (Quran 25:32)
This verse answers your question.
In my opinion, there is no reason to believe that
...compilers of the Qu'uran put a Medinian verse in that of a Meccan Chapter or vice-versa?
One major claim that Quran makes is
15:9 Indeed, it is We who sent down the Qur'an and indeed, We will be ITS GUARDIAN.
If Allah is supposed to gaurd this book then NO ONE has the right to change either the content or the order of the verses.
One thing that many people seem to think is that Quran is a compilation of self contained verses. This is not true. Quran is a coherent book, every verse has specific meaning in the cotext of verses before and after it, and even the surahs seem to be connected to one another.
Here is an interesting study that I came across recently:
Structure and Qur'anic Interpretation: A Study of Symmetry and Coherence in Islam's Holy Text (Islamic Encounter Series) Paperback – September 16, 2014 by Raymond Farrin (Author)