Most people, at least in these latitudes, will instinctively understand the difference between "I could" and the less certain "I might could," or even the upbeat "I just might could," which has a ring of positive acquiescence to it. But there's usually no need to articulate these linguistic distinctions - except of course to folks who, as they say in Charleston, come from off.
But if our critic must have some official authority for "might could," which in grammatical circles is known as a double modal, he could consult any authoritative guide to English usage, including Southern regional usage. The experts may refer to the double modal as informal or conversational, but that scarcely makes it less useful. It's certainly more nuanced.
To quote one linguist, "modal forms such as 'could' and 'should' are ambiguous in Modern English, as they have both an indicative and a subjunctive sense. The use of double modals in Southern American English fills a gap in Standard English grammar, namely the loss of inflectional distinction in English between indicative and subjunctive modals. Dialect or regional forms are often more progressive in gap-filling than is a standard language." Which I hope is a sufficiently technical explanation to appease our critic.
Perhaps the most common example of the greater precision of Suthuhn as opposed to Standard American Usage is the pronouny'all, the second person plural. The less discerning standard usage has only you for both plural and singular. Talk about a linguistic gap that needs filling.
I rest my case, y'all.