The books of the Septuagint (= LXX, here not the Septuagint "proper", which is limited to the Pentateuch, but the whole of the Jewish scriptures in Greek) were produced by different translators; the various books thus exhibit vastly different styles and approaches to the task. LXX-Proverbs is well known for being among the most "free" in making the Hebrew text* available for Greek readers in antiquity.
* For the sake of this answer, I'll take the masoretic-text (MT) as the basis of comparison, although I'm not thereby pre-judging the question as to what Hebrew the Greek translator might have had in front of him.
The "special" problems of LXX-Proverbs are immediately apparent to anyone who reads the Greek beside the Hebrew. LXX-Proverbs already attracted specialist attention in the work of the seminal scholar of the 19th Century, Paul de Lagarde, in his Anmerkungen zur griechischen Übersetzung: Der Proverbien (Brockhaus, 1863). A general orientation to the ancient versions of Proverbs is provided by R.N. Whybray, The Book of the Proverbs: A Survey of Modern Study (Brill, 1995), chapter 7, "Texts and Versions".
The problem here in giving a straightforward "yes" or "no" answer to this way of putting OP's question...
Is there any evidence to suggest that the translation of the LXX was based on a Hebrew text different from the MT?
...is that there is an abundance of evidence -- but not quite of the right kind (no "smoking gun"). There are, however, basically two options.
(1) The Greek translator also acted as "editor"
For some, the many, many small scale differences between the LXX and MT are most readily explained by the translator's evident penchant for "Hellenizing" his proverb collection. Here's what Jan de Waard has to say on this by way of summary to his section on the LXX in his edition of Proverbs for the Biblica Hebraica Quinta:
It should particularly be stated that few Greek translators have been so much receptor language and culture oriented as the Greek translator of Proverbs. His respect of the rhetorical canons of Greek poetry is remarkable. ... This insight will certainly put some restrictions on the relevance of G[reek Proverbs] for Hebrew textual criticism.1
To de Waard's work may be added at least two others. First, this view was anticipated long ago by Henry St John Thackeray (best known today for his Loeb translations of Josephus). In his Schweich Lectures published in 1921, he characterized our translator as a "classical scholar" who "happily, put much of his work into verse".2 More recently, the industrious Johann Cook has contributed many studies arguing for the editorial nature of the work of the translator of LXX-Proverbs. Both his monograph on LXX Proverbs, as well as the introduction to his translation for NETS bear this out at great length.3
It isn't surprising, then, that this line of thinking would lead to the conclusion that the large-scale differences, too, find their origins in the hands of this creative, Hellenizing translator.
(2) There was a different Hebrew edition in antiquity
However, as mentioned above, there is no "smoking gun". In the case of other texts -- most notably Jeremiah -- there is a similar relation between MT and LXX, but for which there is Hebrew evidence of the LXX-like text from among the Dead Sea Scrolls. Naturally, then, this raises the possibility of different Hebrew "recensions" (or editions) in antiquity. For some, like Jeremiah, we have the good fortune of accidental preservation of evidence. For others, we don't.
There have been advocates for differing Hebrew recensions in antiquity, which thus gave rise to, and are reflected in, the distinctive shape of LXX-Proverbs -- some "micro", but some "macro" too, as in the examples from OP. Foremost of these scholars is Emanuel Tov. He registers this possibility ("Two Parallel editions of Proverbs") in his standard work, Textual Criticism of the Hebrew Bible (3rd edn; Fortress, 2012), p. 304; he argues for it at length in an earlier essay, originally published in 1990, but more conveniently available among his collected essays.4
Tov noticed, of course, the clear precedent of Jeremiah, but also the fact that many of the differences between LXX and MT are not of the Hellenizing variety, but rather have to do with loose structure, missing or differing headings, and so on. That is, these are differences which are more readily explained by a different Hebrew edition, than by the translator's obvious fondness for Greek language and culture.
In this he was anticipated by H.B. Swete.5 Referring specifically to the macro-reordering stated in the Question, Swete writes:
Evidently the order of this portion of the book had not been finally settled when the Alexandrian translator did his work.
Tov also has notably been followed by Michael Fox (author of the two-volume Yale Anchor Bible commentary on Proverbs) who is preparing the Hebrew text for what was called the Oxford Hebrew Bible project. In an essay reflecting on this task,6 he makes explicit his agreement with Tov, and adds:
More precisely, the LXX is a translation of a recension, that is to say, one that descends from a deliberately reworked Hebrew text, and his not merely a translation of a corrupted and elaborated copy of the MT.
That such informed, able, and judicious specialists align on different sides of this question should obviously counsel caution. (I put that in bold for a reason...)
Meanwhile, my own inclination (FWIW) is towards the latter position (Tov/Fox). They are, I believe, quite right to note that the differences between MT and LXX Proverbs are not all of one kind, and the difference between the differences (!) is best explained by positing different Hebrew recensions in antiquity, one unlike our MT being the basis for the classical-loving Greek translator that took on this project in the early centuries BCE.
- J. de Waard, Proverbs (Biblia Hebraica Quinta, 17; Deutsche Bibelgesllschaft, 2008), p. 8*. See also his specialist study, “Some Unusual Translation Techniques Employed by the Greek Translator of Proverbs”, in R. Sollamo and S. Sipilä (eds), Helsinki Perspectives on the Translation Technique of the Septuagint (Helsinki and Göttingen: Finnish Exegetical Society and Vandenhoeck & Ruprecht, 2001), pp. 185-193.
- H. St-J. Thackeray, The Septuagint and Jewish Worship: A Study in Origins (London, 1921), p. 13 (he gives some examples).
- J. Cook, The Septuagint of Proverbs: Jewish And/or Hellenistic Proverbs? Concerning the Hellenistic Colouring of LXX Proverbs (Brill, 1997), which has had mixed reviews at the level of detail; see also the PDF of his NETS Proverbs.
- E. Tov, [PDF] = "The Recensional Differences between the Masoretic Text and the Septuagint of Proverbs", in The Greek and Hebrew Bible: Collected Essays on the Septuagint (Brill, 1999), pp. 419-431.
- H.B. Swete, An Introduction to the Old Testament in Greek (2nd edn; Cambridge, 1914), quote on p. 241.
- M. Fox, [PDF] = "Editing Proverbs: The Challenge of the Oxford Hebrew Bible", Journal of Northwest Semitic Languages 32/1 (2006): 1-22; quote on p. 4.