"Let's you and I/me be fair with each other."
'Let's you and [I/me] be fair with each other.'
'Let's you and [I/me] indulge in a little bit of reverie.'
Should "I" or "me" be used in these two sentences, and why?
Given that Let's is a contraction of Let us, if you want to be redundant and expand upon just who the (contracted) "us" consists of, then since "us" is the objective case, the people included in "us" should also be mentioned with the objective case: you and me.
Without the contraction, does "Let I be fair with you" or "Let we indulge in a little bit of reverie" sound correct? (I certainly hope not.)
If you're wondering where the subject is in sentences that use Let's, keep in mind that it is an imperative verb: the subject is always implicitly the person you are speaking to.
I'm not much interested in "correct" on this one - I wouldn't like to hear the Queen talking about "Me and my husband", but otherwise I'm fairly relaxed about such things.
As @Janus comments, the whole construction is somewhat colloquial/dialectal/idiomatic. But overall prevalence has been pretty evenly-matched for generations. Personally, I'm happy with both versions - I'd usually use me, but I might use I sometimes in [mock-]formal contexts.
On the one hand, the standard two-person equivalent of "us" (as in "let us," which in contracted form yields "let's"), when the two people involved are the speaker and the person spoken to, is "you and me"; whereas the standard equivalent of "we" under the same conditions is "you and I," This would lead to the usual choice of "you and me" in the two examples you list above.
On the other hand, we have the famous example of the opening lines of T. S. Eliot's "The Love Song of J. Alfred Prufrock":
Let us go then, you and I,
When the evening is spread out against the sky
Like a patient etherized upon a table;
Let us go through certain half-deserted streets ...
It seems to me that the appositive "you and me" or "you and I" serves in these examples not merely as a redundant expression of the already identified "us" in each of these instances, but as a way of specifying the limits of that "us": "let us be fair, you and me/I," "let us indulge, you and me/I," "let us go then, you and me/I," as opposed to, say, "let us X, you and me/I and everyone we know"—which perhaps loosens the usually tight equation of "us" = "you and me."
Bryan Garner, Garner's Modern American Usage, third edition (Oxford University Press, 2009) provides the following discussion of the expression:
Let's you and I. First, think of let's: let us. Us is in the objective case. Another form of the phrase (still in the objective case) would be let you and me (you and me agreeing with us). The construction let you and I is ungrammatical—and fairly rare.
But what about let's you and I? This, too, is ungrammatical—us and you and I being in apposition. [Cross reference omitted.] It's an error of some literary standing. T.S. Eliot began "The Love Song of J. Alfred Prufrock" (1917) in this way: "Let us go then, you and I." In that sentence, go is an infinitive without an express to (sometimes called a "bare infinitive"), and an infinitive has as its subject a pronoun (us) in the objective case—not in the nominative case. Yet the appositive for us—namely, you and I—is in the nominative case. This is an oddity, but today's let's you and I [+ verb] is common in spoken and written English alike. H.W. Fowler would have called it a "sturdy indefensible" [examples omitted].
As Garner says, "let's you and I" is by no means unusual in written dialogue. From Sam Porter Jones, "Unload Your Hearts," in Sermons by Rev. Sam. P. Jones, as Stenographically Reported (1887):
Brother! Young man! Father! Husband! Hear me a minute now.Let's you and I help unload mother's heart tonight! Let's you and I help unload wife's heart to-night! Let's you and I help unload our children's hearts to-night.
From Dorothy Parker, "Just a Little One" (New Yorker, May 12, 1928), reprinted in Complete Stories (2002):
I'd like to take care of it and comb its hair and everything. Ah, don't be stuffy about it, Fred, please don't. I need a horse, honestly I do. Wouldn't you like one? It would be so sweet and kind. Let's have a drink and then let's you and I go out and get a horsie, Freddie—just a little one."
And to represent the many, many more-recent instances, from Thomas Bruce, I'm Still a Young Man (2008):
“Let's you and I throw a set Saturday night.” I said “hey brother, I don't have any dust.” “Man, it don't take a whole lot of money, just something small.” “We'll invite a few dames over.” “Let's you and I try to have a party at least once a month."
A lengthy (and only partly reproduced online) descriptivist discussion of "let's you and I" versus "let's you and me" appears in Raphael Salkie, Pierre Busuttil, and Johan van der Auwera, Modality in English: Theory and Description (2009).