While I haven't seen your Liverpool, my 4" - 4" 1/2" Liverpools have thicker mouthpieces than my 3 1/2" snaffles. They are made for bigger mouths. I haven't seen a Liverpool smaller than 4". While there is such a thing as too big, a thicker bit is generally more kind than a thin one, within reason. It would be the difference between rubbing dental floss back and forth on your tongue verses a clothesline cord. You are dispersing the pressure of the bit across more surface area. A bradoon snaffle is a more severe bit than a hollow mouth eggbutt. Of course, if they can't close their mouth around it, it is obviously too thick.
Anytime you start miniaturizing something, smaller increments matter. We have to use different harnesses on our A-size horses than our B-size horses, with just four inches between them. That same four inches in 15 and 16 hands is not a problem. I don't think the mini bit industry is fully stocked yet. They need someone to produce appropriate sized bits reasonably priced, but they know that generally mini horse people won't spend as much as "big" horse people, with exception to those that get into Mylers. Show me a mini driver that has spent $4K+ on a vehicle. It's not necessarily uncommon in the big horse carriage/CDE world. And considering that the majority of the cost of a vehicle is NOT in parts, but craftsmanship, just because vehicles are small doesn't mean they should necessarily cost less. Same goes for bits. My best mini/pony bit was made in England for $100. My point is that if people want better quality, they are going to have to spend more.