"A might bit" or "a mite bit"?

From Vocabulary.com, a mite

is one of many words for a very small amount.

The first definition in the OED for mite refers to money, with a quote from the 1700s calling a mite about 1/3 of a farthing. The widow's mite is a well known expression for a charitable contribution made by a poor widow.

Definition #3 in the OED (same link) is a

a very small amount

There are numerous quotes in the OED, the most recent of which is:

1993 T. Hawkins Pepper xii. 248 All evening he's seemed a mite awkward.

The OED has no entry for mite bit, or might bit, and I did not see mite bit in any of the quotations it gave for mite.

Thus mite is found in both current British and US English, but is somewhat of an affectation. In US English, mite is probably most common in Appalachia. See Daily Yonder, Mountain Talk

In the corner of Appalachia where Tennessee meets Virginia, where this ... can tell what part of Appalachia people come from by the words they use. ... Regardless of where you live in West Virginia, we are all blessed with a bit of that southern twang. ... A-mite — a little You're lookin' a-mite peak-ed today.)

As for mite bit in US English, I found an entry in the Urban Dictionary, which said it is "ironic", as in (quoted very loosely here) a million might be a mite bit high for that house.

I think the phrase mite bit is used mostly either ironically or as part of Southern or Appalachian dialect, but I don't have a reference for mite bit in Appalachian dialect.

It is definitely, absolutely, not might bit.