From the Borrowed Chords theory, I should be able to use the flat-third major (♭III) in my composition and it should sound good. However, after many trial and errors, I am unable to discover any suitable chord progression that utilizes the flat-third major pleasingly. Does anybody know how to apply the flat-third major chord into a song?
It's fairly 'in yer face', and works best in blues and soul. Think 'Knock on Wood', often used between I and IV. Good as a turnaround , I-♭III- II- ♭II -I, first four in the last bar of sequence.
You say 'I can use'. You can use anything you like in your songs - they don't have to obey any rules or theory. They may well do, but they don't have to. If it sounds good...
In a minor key, you can use: i - ♭III - ♭VII - IV. Since this is minor key, the flats are redundant, but I like to include them anyway, for clarity. Several songs use this progression, but one that comes to my mind is "Radioactive" by Imagine Dragons (Bm-D-A-E). I'm not at a keyboard, but I'd imagine the same progression would work reasonably well if you replaced i with I.
From a functional harmony perspective, ♭III seems to have a tonic function.
The basic classical theory is that borrowed chords have the same function as the diatonic chords for which they substitute. So, a borrowed minor
iv still functions as a subdominant.
But you might consider a different theory if you are dealing with rock music. Some rock music used chord roots based on a minor scale, but chromatically makes the chord qualities on those roots major. In other words, roots
♭III come from the minor orientation of a lot of rock music, but chords are harmonized with major thirds. It's a bit similar to how blues adds minor sevenths to
I IV V even though those tones aren't in the "key."
This post discusses the style with one textbook source calling in "chromatic minor": Harmonic succession in the chromatic-minor system
From this perspective
♭III isn't borrowed. It's part of a different, unique tonality.
Use the ♭VII chord as well. That nice fourth interval between ♭III and ♭VII helps to move there nicely, or at least to set it up tonally, thirds can be a nasty jump. Jarring if nothing else, so nice subdominant to tonic, IV - I or related ♭VII / ii - I, can really smooth that out. Also minor iv.