Overcheck bits are not commonly used with miniatures, mostly because their mouths can be too small to support two bits between the bars.
Personally, with sidecheck bits especially, I don't usually find a need to use a separate bit. You CAN hook your check (either side or over) directly to the perch of the driving bit or onto a snaffle ring.
Separate check bits are useful for horses that have trouble keeping their heads up, and separate check bits are almost ALWAYS used exclusively for horses that are checked up super high/hard (usually show ring only). They allow the horse to not have confusion between the operation of the check bit and the driving bit, as the check can interfere with the operation and mechanics of a driving bit if the horse is checked up very high (it pulls on the bit differently than if the driving bit was laying in the mouth.)
Some horses need more severe overcheck bits if they lean hard on the check. The most severe would probably be the twisted wire ones you saw. The least severe ones tend to be the mullen type ones, or the leather ones, but remember the bits are very slender like a bradoon and most are fairly "harsh" but it is only the horse itself that applies any pressure and not really the driver's hand. If you do use a severe one it is extremely important that you properly condition the horse's neck muscles to carry the head and neck that high, otherwise it is unfair to ask the horse to work in such extreme discomfort.
If you have a horse or mini with a very small mouth that may not be able to support two bits, you can consider using some black shoelace or cord. I prefer a strong cord, as it is a little less thin and severe than a shoelace, but again it totally depends on your horse and how it is trained.
Like any bit, you will find your horse will prefer or work better in one bit or another.
I hope this helps any,
Andrea