Whether to seek correction after my paper is published when the order of my first, middle, and last names is incorrect?
I can relate to the discomfort of having your name misspelled, however, I don't see how to completely resolve the issue.

First, if the mistake was already present in the proofs, then you're in fault for not noticing and there is nothing to be done. If the mistake was introduced after the proofs were accepted by the authors, then it's the publisher's mistake. But to my experience, there's little they can do to remedy the situation.

As anedoctal evidence to this, I recently had a very respected publisher re-order our list of authors after the proofs (second and third authors were switched in a four authored paper - order was initially alphabetical). The publisher offered to publish an errata, that is, a minor one liner in a forthcomng issue stating the correct author order of our paper.

Although not as reconforting as having a correction in the actual paper, it's not completely useless as you can prove (in your case) that it was actually a mistake and you are not simply claiming a random paper in your CV, but I don't think you will ever need to prove this. The issue is that your paper will still be published and cited with the incorrect name, and probably not correctly indexed e.g. in your Google Scholar profile.

TL;DR contact the publisher and ask for a correction, but expect no more than a published errata.