The numbers that define the designation of the gearboxes are not just random numbers plucked from the air, they are supposed to be the linear distance between the centerline of the input shaft & the cluster shaft.

The 180B SSS gearbox is a 63 series which means that the centerlines are 63mm apart. To fit this into a 56 series case is going to take a lot more than some "mild machining to fit the bearings in"
For what it's worth, all of the 3 speed, 4 speed & 5 speed 56A gearbox cases are the same. The difference is in the sandwich plate & the extention housing, but the gearbox case always has 4 gears inside & any number above that is housed in the extention housing.

I don't know where you 'heard' about the big gears in little case conversion, but you can tell the informant that it would be wise if he confirmed the info before passing on yet another Furphy.
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[For overseas readers, Furphy is Aussie for "misinformation", a word that goes back to the trenches of WW1 when the drivers of the watercart [made by Furphy] would pass info along the line on their rounds. Invariably, with each re-telling, the info became less factual or current]