Strips of birch plywood will be glued and stacked between the studs while leaving open channels for the studs and shafts. The diagram shows the basic idea without particular reference to scale other than the base image of the flange bearing as a guide. The square void in the center could be reduced further provided the shaft does not rub on the plywood. Quadrant sections could fill the corners but would not provide much extra strength being so close to the axis.
Further layers of plywood could easily be added to the outside of the structure shown above but would not contribute to compression resistance unless the spreader plates were made oversized to match. Where does one draw the line? Plywood is quite cheap so adding layer after layer is easy enough. Does extra bulk add anything to the overall stiffness of the "box" structure? Being so far from the axis it should have considerable extra 'moment' over the inner layers for structural gains.
The image shows the basic dimensions of the inner face of the flange. If it were left bare [i.e. without a load spreading plate] the plywood edges would be crushed first where the flange is machined flat. But left completely untouched where the casting moulding is recessed. The recesses could be filled to spread the compression loads more evenly. A "rubbing" on paper could be used as a pattern for thin plywood to increase the active surface area for very little effort.
It looks from the flange dimensions as if the required thickness can be made from three layers of 15mmm birch plywood. These would just reach the outer limits of the bearing flanges and just clear the shaft. If four layers of 12mm plywood were preferred then localized 'woodworking' could solve any shaft or bearing clearance issues.
Making the overall structure any larger demands oversized end plates. Though I am trying to resist having 'naked' edges to the plywood end plates. It would look much neater to have the sides overlapping the end plates for a more uniform appearance but completely ignoring compression resistance. A thin, aluminium "wrap" would easily enhance a roughly finished plywood block if an easy cosmetic skin is desired. Far easier than trying to sand the 'lump' perfectly smooth and square all over!
The entire structure could be fitted inside an 8" [PVC drainage] pipe for weather protection and an even smoother appearance. Drainage pipes have standard end caps which could protect the bearings with only a suitable center opening cut for the axis shaft to exit. The plywood structure could even be built up in extra layers. Perhaps until the plywood cylinder completely filled the 8" pipe if this was thought desirable for structural reasons.
The images show the simple process of making a paper pattern by 'rubbing.' The pattern was then cut out and glued to the 6mm thick, flange packing material. In this case a cheap plastic cutting board as I had no 6mm plywood handy. I then sawed around the pattern with a hand fretsaw with coarse omni-directional blade. The power jigsaw immediately welded its blade to the plastic regardless of using a very low speed or choice of metal or wood cutting or oil.
A little filing to "tidy up" the edges resulted in a solid surface instead of a large recessed area. The packing piece will never be seen so absolute accuracy was not important. [Interestingly[?] the pattern was handed and could not be rotated 180°] I think you will agree that the packing increases the flange's active area considerably and will help to spread the compression forces more evenly. Anyone who cares can overlay a grid to calculate the exact increase in area. At a glance, I'd say the area has nearly doubled. All of which helps to protect the end grain of the plywood laminations under compression.
The diagonal measurement of these particular 50mm flange bearings is 188mm. This will just fit inside an SN4 200mm PVC pipe. But not the SN8 because of its greater wall thickness. Variations in flanges for different duty loads and by different manufacturers are available. A pipe would set the plywood size limit on a round form. Otherwise the corners would extend well beyond the near 8" diagonal of the bearing flanges unless rounded off. While a thin, aluminium covering can be any size and shape you choose.