In a typical Eastern Catholic fashion, I consider the Eastern and Western approaches to be reconcilable, and I maintain the Eastern approach while being in communion with Rome. I will approach this question from different perspectives.
Linguistic perspective
The original language of the creed was Greek. In Greek, the term proceed has a very specific meaning: the thing that proceeds from a source takes its eternal origin from that source. This is why the Eastern Churches insist that the Holy Spirit takes its origin from the Father alone, or at most from the Father through the Son. In Latin, however, the term proceed has a broader definition and can include an intermediary source, which is why Latins have no difficulty in speaking of a double procession while maintaining that the Father is the principle without principle, the Latin equivalent of the Greek assertion that the Father is the cause.
Patristic perspective
It is often claimed, by Orthodox, that Rome altered the apostolic faith by changing the creed. I am not at all convinced by this argument, for a simple reason. While it is true that Rome changed the creed, the Filioque was not at all an innovation in the West. Virtually all Western Fathers professed a double-procession since at least the 4th century. Therefore, demanding that Latins reject the Filioque would amount to them throwing away their patristic tradition altogether.
Theological perspective
Now, what is meant by a double-procession? Latins are often accused of making the Son a cause of the Holy Spirit, or of introducing a second cause in the Holy Trinity. However, that is not what the Roman Church teaches. The Roman Church teaches that the Holy Spirit proceeds from the Father and the Son as from a single source and maintains that the Father is the principle without principle. It is therefore not true that a second cause is introduced in the Trinity. The Father remains the sole cause of the Trinity, and from Him, and alongside Him from the Son, proceeds the Holy Spirit - and, as we have seen, the term proceed has a broader meaning in Latin, which permits this definition. And to show that this is not simply arguing in bad faith, the Roman Church forbids its Latin members, who live in Greece and who pray in Greek, of adding the Filioque to the creed, because the Roman Church agrees that, in the Greek language, a double-procession would indeed be heretical.
Conclusion
These are my humble thoughts, but keep in mind that I am not a theologian and, more importantly, I am not well versed in Western theology. Allow me, however, to recommend two key readings to get up-to-date on that issue.
1- Greek and Latin Traditions Regarding the Procession of the Holy Spirit. This document, authored by the Pontifical Council for Promoting Christian Unity, can be found here: https://www.ewtn.com/catholicism/library/greek-and-latin-traditions-regarding-the-procession-of-the-holy-spirit-2349
2- The Filioque: History of a Doctrinal Controversy. This book, authored by an Orthodox scholar, can be found here: https://www.amazon.com/Filioque-Doctrinal-Controversy-Historical-Theology/dp/0195372042
Thank you for your very good question!
Off the top of my head:
The Filioque: a Very Basic Introduction is a good short overview.
There is Siecienski's book, however keep in mind he is Eastern Orthodox and the book is biased because of that.
I've heard Oberdorfer's book on the Filioque is much better, however it's only in German.
You can check out this history of the doctrine by and old Anglican scholar H. B. Swete.
Here's a basic video summary of some Catholic arguments for this doctrine.
Eh, thank you, but none of it is mine. It's all Siecienski's. You can (and should) buy his two works:
https://www.amazon.com/Filioque-Doctrinal-Controversy-Historical-Theology/dp/0195372042
https://www.amazon.com/Papacy-Orthodox-Sources-Historical-Theology/dp/0190245255/ref=asap_bc?ie=UTF8