Setting aside contemporary, polysemous uses of 'to bend the rule' (and variants), and retreating to the earliest known uses of the idiom, which appeared in the same century as the earliest known figurative uses of the related 'Lesbian rule', it appears that the idiom 'to bend the rule' was originally associated with the figurative 'Lesbian rule'.
The earliest known uses of 'to bend the rule' appear in religious literature of the 1600s (1634, 1669; 1689 OED), that is, subsequent to and, as will appear, concurrent with the earliest known figurative uses of 'Lesbian rule' (1605, a1628 OED; 1630). The association of the two is by straightforward textual parallelism of the idiom at its first known appearance, and in the form of a noun phrase, with the figuratively used noun phrase 'Lesbian rule'.
A digression into the lexical aspect of 'Lesbian rule' supplies the chronological and conceptual background for its association with the origin of 'bend the rule'. OED Online gives this definition of the figurative use of 'Lesbian rule':
Lesbian rule n....fig., a principle of judgement that is pliant and accommodating. (Very common in 17th c., but app. not always correctly understood.)
["Lesbian, adj. and n.". OED Online. December 2016. Oxford University Press. http://www.oed.com/view/Entry/107453?redirectedFrom=lesbian+rule (accessed January 05, 2017).]
The first OED attestation of 'Lesbian rule', in the form "Lesbian square", appears to be a concrete reference used to leverage a poetic metaphor:
1601 S. Daniel Epist. to Sir T. Egerton 131 That Lesbian square, that building fit, Plies to the worke, not forc'th the worke to it.
The second use attested in the OED is certainly figurative:
1605 T. Tymme tr. J. Du Chesne Pract. Chymicall & Hermeticall Physicke II. ii. 111 The composition and wonderful nature thereof is, as it were, a certaine example and Lesbian rule of our worke.
The third use attested (in the variant form "Lesbian Squire"), from 1606, is at least on its face a literal use referring to an architectural process. The fourth attestation, however, is of greater interest in connection with the question at hand:
a 1628 J. Preston New Covenant (1630) 233 Thou goest not by a straight rule, but by a leaden Lesbian rule.
The reason this use is given as ante 1628, with the publication date at 1630, is that John Preston is known to have died in 1628; the sermon containing the quote, "The Seventh Sermon" (of fourteen), was delivered prior to that date.
Digging behind the scenes, into a 1634 edition of Preston's New Covenant, it appears the full quote is this:
...now thou goest not by a straight rule, but by a leaden Lesbian rule, by a bended rule.
The New Covenant; or, the saints portion (1634), p. 233, John Preston.
This shows the straightforward textual parallelism associating the phrase 'Lesbian rule' and the idiom 'to bend the rule'. It is also the earliest example of the latter I found, and is applied in a noun phrase, rather than as a verb.
I will provide attestation of other early uses of the idiom 'to bend the rule', the dates of which uses dovetail with figurative uses of 'Lesbian rule' in the 1600s (as listed parenthetically for both phrases in paragraph two of this answer), although the uses provide no further direct textual evidence of the association of the two.
First, though, here is another early (1630) instance of 'Lesbian rule', not shown in OED Online, from 17th century religious literature:
... Particlar Churches have erred; therefore the best security from error is in the Scriptures.
This is a Lesbian rule, able to decide all Controversies, and it is vitio hominum, by the fault of bad Interpreters that it doth not. ....
The workes of Thomas Adams (1630), p. 558.
OED Online attests the date of the first use of 'to bend the rule' as 1689:
f. to bend (also stretch) the rules: to interpret the rules leniently; to overlook or allow an infringement of the rules.
1689 R. Atkyns Enq. Power Dispensing with Penal Statutes 36 Let us not..approve of all things, tho' delivered by Authors of greatest Name, for they often serve the Times, or their Affections, and bend the Rules as occasion requires.
["rule, n.1". OED Online. December 2016. Oxford University Press. http://www.oed.com/view/Entry/168717?redirectedFrom=bend+the+rules (accessed January 05, 2017).]
Two earlier uses in religious literature of the 1600s, in addition to the earliest use (shown in the foregoing), chronologically support the original textual association of the idiom with 'Lesbian rule'.
One is from another of John Preston's works, also published in 1634 but, just as with New Covenant, composed prior to 1628:
Therefore hence observe, the God must needes be most holy, and righteous, and just in all his wayes; because there is neither love, nor hatred, nor griefe in him, nor joy, which should make crooked or bend the rule of his will, or alter it in any action.
Life eternall; or, A treatise of the knowledge of the divine essence and attributes (1634), John Preston. Bold emphasis mine.
The other is this from the 1669 An elegant and learned discourse of the light of nature, p. 140, by Nathanael Culverwel:
... Though they be not come to such an height as this; yet, either by their flat, and frigid explicating, they do endeavour to dispirit, and enervate the Word of God; or else, in a more violent, and injurious manner, they do even ravish it, and deflower the virginity of it; or else in a more subtle, and serpentine manner, they seek to bend the Rule, and expound it to their purposes, and advantages. The Letter of the Word, the vagina verbi, that does not wound them, that does not strike them: and as for the edge, they think they can draw that, as they please; they can blunt it, as they list; they can order it, as they will.