Fido. A few thoughts about crank shims.
Longstroke Crank has a drive side ball race BSA Part 67-1240. This is an Imperial size bearing, Shaft 1 1/8", OD 2 13/16" Width 13/16" The later roller type bearing is a different size, 30x62x16 mm. They are different shaft sizes but if considered necessary the shims from the later engine would just about do. Longstroke shaft is 28.575mm, very slightly smaller than the later cranks and I can't see crank shims listed in the BSA Longstoke parts section.
Tightening the cush drive on a Longstroke locks the bearing against the crank as usual and as the bearing is a deep ball race type, there is no float other than that manufactured tolerance within the bearing. So looks as if crank shimming is not required to the precision of the later design, featuring a single lipped roller bearing allowing float. Suggested limit before tightening everything up is 5-10 Thou (0.1-0.3mm) for the Longstroke crank. In my estimation until the cush is tightened the shaft must move easily within the inner race.....I doubt this happens in practice and the simple answer is to replace any exiting shims.
BSA missed a trick with the ball race design here, as there is nothing to hold the outer race into the crankcase other than the press fit. In other words friction and luck. The later design does not put a sideways (axial) load towards the timing side onto the outer race, which would tend to draw the whole bearing from the crankcase as is the case with the deep groove race.
Ball Race 67 1240 is omitted from BSA Service Sheets, so not readily cross referenced to commercial brands but from the size shown a replacement can be found if required.. C3 Grade usually used in this operating situation if a choice is offered.
The pistons you have with a BSA stamp should also have reference number. There is a piston chart in the Longstroke Parts book, 1949, in the Forum Literature section and the piston can be identified. Thanks to JulianS for originally posting this valuable reference.
Old stock original rings are a better choice than their more modern copies. Original Hepolite, COVMO or Milemaster were the favoured brands when the bikes were in daily use and will fit straight from the pack. Ebay is a good source.
Swarfy.