In order for the filter to operate in ideal conditions, the filter is meant to be held stationary in an inverted position. You can buy mounting brackets for the filter housing, but I didn't have any available so I would need to make one. I was able to find a perfect-sized chunk of aluminum angle material which would do the job if mounted to the top of the compressor, making a nice contained setup.
The aluminum bracket requires two main cuts: one square cut for the air fitting which would hold the filter, then a pair of holes which would mount the bracket onto the compressor. For the square cut, it was a simple imprecise shape that needed only be large enough to hold the 3/4" wide brass air fitting. I cut the shape with my manual Bridgeport mill, using the "mouse ear" method to open the corners up:
The other cut is a pair of holes which would attach directly to the top of the Shoebox's air manifold. The air manifold attaches to the red sheet metal using a pair of #10-32 screws, so I used a set of longer screws to hold the bracket on top.
I plumbed the filter using some spare 1/4" macroline hose fittings. The macroline hose only needs to hold itself between the filter and shoebox, and is only pressurized to the "shop air" around 100-psi.
The new filter worked out as intended and doesn't require much cleaning. Adding insurance measures like this only makes the system more reliable.