Dave, I've never made formal measurements, but the dimensions are visually a match to Bart soapbars. The Super 4 has smaller depth with a radiused top though, so depth could be an issue. I never knew the name of the pickup. The later Musician pickup I got off eBay later just said "Musician Bass Pickup" on the blister card and had a part number. I'm sure I kept the packaging, but it's buried in my garage somewhere. I would imagine that later pickup has a different name since it has a double set of adjustable double polepieces, almost 20 years before Fender started making the similar looking stack-paired Jazz MM-sized humbucker but without pole adjustments for the active USA Deluxe series.
Many of my basses are still at my parents' as my ongoing health issues have kept me from being able to get things straightened up here at my home to bring them here. I was visiting last week and went through the cases and played the ones still there for my 2 year old nephew, whom they keep for my brother during the day. He was enthralled, and I got to complement myself on having some VERY nice instruments. The necks on my G&L L5500 and my Fender Diamond Anniversary Jazz are so unbelievably fun, and the Musician, which is fretless, just sings. I played a bunch of 'music store goof off licks,' complete with crazy slapping, something I never use in bands, but I have developed a few licks I like. He just giggled, danced and clapped. It was a good day.