Paper was rejected for high similarity with parts of my own dissertation
Most publishers I know explicitly permit (pre)publication of papers as part of a thesis. For example the copyright policy of Elsevier (the publisher you used as a tag) states:

Authors can use their articles, in full or in part, for […] [i]nclusion in a thesis or dissertation (provided that this is not to be published commercially)

Going by this, everything is fine with your submission.
More generally, you did not attempt to sell a work previously published in another journal (or parts thereof) as a new publication, which would be the typical case of self-plagiarism.
For all of this, it does not matter whether you modified the respective chapter or included parts of the frame, as long as you stayed away from the already published chapters.
However, the rejection of your paper may not have taken this into account, either because it was automatic or somebody did not connect the dots.
I thus advise to consider the rejection to be an unintentional error or their side.
I would therefore simply write to the journal explaining that you suspect that your paper was mistakenly identified as plagiarism due to similarity with your thesis, which is fine as per the above rules.
In general, it is a good idea to mention upon submission that parts of your paper have been previously published, be it as a preprint or thesis to avoid exactly this as well as potential clashes with journal policy.
Finally, to address some of your concerns:

I feel like [writing to the editor] may be conceived as begging for permission to be allowed into a reviewing process

I would not worry about this.
First, what you are challenging is not a judgement call (e.g., a desk rejection due to lack of relevance).
The journal stated a very objective argument for rejection (plagiarism) and you have very objective arguments that this argument is erroneous.
It’s a simple mistake; it can happen; you are asking for it to be fixed.
Furthermore, even if this should be considered begging, you have little to lose: Right now the journal considers you a plagiarist (which you might want to set straight even if you decide to submit to another journal).

The copyrights of Elsevier allow me to use articles published under their flag in my thesis. My case, however, is the other way round: I want to publish something that has been part of my thesis in very similar form. The thesis is published, but it contains a part that is not peer-reviewed and has not been published as an article.

Copyright transfer to journal exists because the journal’s business model is selling your paper and they want to have some exclusivity on this.
For this, the order of events doesn’t matter.
My point with linking these policies is to demonstrate that Elsevier is generally fine with papers that are available as a thesis.