"If you'll kindly give me a chance to explain" seems to capture the essence of the use you have in mind. It's got a bit of attitude. The particular gem you cite, which sounds ever so Southern, is used in ways both innocuous and not, but more generally the latter, but, when the former, represents an assertion more so than an expression of thanks. In that case it's a way of saying [to the person being thanked] that they're right on the edge of behaving in a way that's objectionable, and, as such, it really is a form of "Please". In its more aggressive form it's rendered "I'll thank you to kindly ..." and then, of course, finished off with whatever's being objected to. And bypassed altogether it's certain to yield something akin to Herman Cain's now famous assertion, "Excuse me!", when plagued by the press of journalists.
[And then, next day]
To properly respond to the author-of-the-question’s comments:
Taking into consideration the context in which I've always heard this, if you'll kindly give me a chance to explain does not capture the essence of thank you kindly. Also, "right on the edge of behaving in a way that's objectionable" does not relate to thank you kindly at all; it's a completely different sentiment. I've always heard thank you kindly as a replacement for thank you after someone does a favor. – Daniel δ
You stated, Daniel δ, that from your perspective the reported parallel between the word kindly and the word please “doesn’t make sense” within, that is, the particular phrase you introduced. I offered you my experience of how the word kindly is used in, mind you, a transparently good faith attempt to provide you a clear means by which to understand that yes, indeed, it is intended to mean please in something of a qualified sense. The clause I started with, “If you’ll kindly give me a chance to explain,” is virtually indistinguishable from the alternate clause “If you’ll please give me a chance to explain”. (You seem to be saying you don’t see that.) Furthermore, I never said it captured the essence of “thank you kindly”, as is your claim here. But I did say that it “seems to capture the essence of the use you have in mind”, to wit, the parallel between the word kindly and the word please. If you’ll simply consider the following extraction from your own original query, “...that kindly can be used as a sort of please. This usage was in my mind when ...,” there’s no question but that I was speaking directly to what you deemed central to your question.
And finally, by stating from the outset that you couldn't quite grasp the legitimacy of replacing please for kindly in the phrase “thank you kindly”, by any reasonable measure you were stating that your experience of the use of the term has yet to provide you the means by which to bridge that gap. So by asking the question you asked, you offered up of a tacit acknowledgement that others had necessarily/likely/possibly experienced the use of that phrase in a way that does bridge that gap. My experience certainly falls within that category, as reported. You have responded by saying that, well, yours does not. We already knew that? We believed you from the door? WTH?
For my part, I must admit, room does exist which allows for confusion to follow from one particular aspect of my contribution. I’ll go in and clean that up. The combination
“and, when the latter, represents …”
“…but, when the former, represents …”,
because from that point forward my focus is on that of [what I deem] the innocuous use of the “thank you kindly” rather than the more common use. Your attention to that failing would have offered of some value. But literally no part of your comment contributes anything in the way of that which is useful or appropriate.
If I’ve incidentally offended you somewhere along the line, I do not know it and did not intend to do so.